1.5 Solving Inequalities
Use Interval Notation
Example 1
Example 2
Solve Inequalities
Example 3
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Example 4
Solving a Combined Inequality
Solve:
Example 5
Example 6
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Example 7
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1.6 Equations and Inequalities Involving Absolute Value
Solve Equations and Inequalities Involving Absolute Value
Example 1
a) |5x| = 15
d)
b) |3 - 8t|-3 = 11
c) |2|x = -16
e)
Example 2
Solve Each Inequalitity. Graph the solution set.
a) |1 - 4x| - 9 < -6
b) |9 - 2x|-2 > 11
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c)
d) |-3x| > |-11|
e) 4|7x| ≥ -4
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2.1 The Distance and Midpoint Formula
Distance Formula:
Midpoint Formula:
Example 1: Find the distance d between the points (-4, 5) and (3, 2).
Example 2
Find the distance between the points (4, -3) and (2, 5).
Example 3: Consider the three points A=(-2, 1), B=(2, 3), and C= (3, 1).
a) Plot each point and form the triangle.
b) Find the length of each side of the triangle.
c) Verify that the triangle is a right triangle.
d) Find the area of the triangle.
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Example 4
If (4, -2) is the endpoint of a line segment, and (-9, -1) is its midpoint, find the other endpoint.
Example 5
Find the midpoint of the line segment from
to
Example 6
A Car and a Truck leave an intersection at the same time. The car heads east at 30mph, while the truck heads south at 40mph. Find an expression for the distance apart
d (in miles) at the end of t hours.
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2.2 Intercepts; Symmetry
Find Intercepts
To find the intercepts of an equation:
x-intercept: Make y=0, solve, (#, 0)
y-intercepts: Make x=0, solve, (0, #)
Example 1: Find the intercept(s) of the graph
Test an Equation for Symmetry
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Example 2: Test
for symmetry.
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Example 3: Test
for symmetry.
Example 4: Given the equation, test for symmetry.
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2.3 Lines
Objective 1: Calculate and Interpret the Slope of a Line
Example 1: Find and interpret the slope of the line containing the points (1, 2) and (5, -3).
Objective 2: Find the Equation of the Line Given Two Points
When you see FIND, EQAUTION, LINE, follow these steps:
1. Find the slope
2. Using slope-intercept form y=mx+b, plug m (the slope), x and y from a given point to get b.
3. Once you have b plug it into y=mx+b along with m (the slope).
4. If needed Standard/General Form: Ax+By=C (fractions are not allowed and A must be positive).
Example 2: Find an equation of the line containing the points (2, 3) and (-4, 5). Express in Standard Form.
- Parallel Lines: Have the same slope, but different y-intercept.
- Perpendicular Lines: Have negative reciprocal slopes, meaning you flip and change the sign. The product of perpendicular slopes = -1
- If the equation is the same, the lines are the same.
Example 3: Find an equation of the line that contains the point (1, -2) and is perpendicular to the line x+3y=6. (b) Find the line containing the same point
and parallel to the same line.
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Undefined Slopes
x=# has always an undefined slope. It creates a vertical line.
Zero Slope
y=# has a slope of 0. It creates a horizontal line.
Example 4
Graph the line that goes through (2, -3) and has an undefined slope. Find the equation of the line.
Example 5
Graph the line that goes through (-1, 9) and has a slope of 0. Find the equation of the line.
Example 6
Find the equation of a line that is parallel to the line x=11 and contains the point (-2, 7).
Example 7
Graph 3x-2y=6
To graph using a ti-84:
1) Press y=
(top left)
2) Make sure the equation has isolated the y and input the equation into y1=
3) Press graph
4) To get points (table), press 2nd, then graph.
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2.4 Circles
General Form of a Circle:
Everything is foiled from standard form.
The Unit Circle: (used in calculus)
Example 1: Write the standard form of the equation of the circle with radius 5 and center (-3, 6).
Example 2: Graph the equation:
Example 3: For the circle
, find the intercepts, if any, of its graph.
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Example 4: Graph the equation:
Example 5: Graph
.
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Example 6
A Ferris wheel has a maximum height of 331 feet and a wheel diameter of 310 feet. Find an equation for the wheel if the center of the wheel is on the y-axis and y represents
the height above the ground.
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2.5 Variation
Varies Directly, Directly Proportional, and Constant of Proportionality
If y varies directly with x,
x > 0, and k, k > 0, is the constant of proportionality, then the graph illustrates the relationship between
y and x.
Note that the constant of proportionality is, in fact, the slope of the line.
Example 1: Mortgage Payments
The monthly payment p on a mortgage varies directly with the amount borrowed B. If the monthly payment on a
30-year mortgage is $6.25 for every $1000 borrowed, find a formula that relates the monthly payment p to the
amount borrowed B for a mortgage with these terms. Then find the monthly payment p when the amount
borrowed B is $200,000.
The graph illustrates the relationship between the monthly payment p and the amount borrowed B.
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Varies Inversely and Inversely Proportional
The graph illustrates the relationship between y and x if y varies inversely with x and k > 0, x > 0.
Example 2: Maximum Weight That Can Be Supported by a Piece of Pine
The maximum weight W that can be safely supported by a 2-inch by 4-inch piece of pine varies inversely with its
length l. Experiments indicate that the maximum weight that a 8-foot-long 2-by-4 piece of pine can support is
400 pounds.
Write a general formula relating the maximum weight W (in pounds) to length l (in feet). Find the maximum
weight W that can be safely supported by a length of 20 feet.
Example : Modeling a Joint Variation: A Student’s GPA
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Example : Modeling a Joint Variation: A Student’s GPA
A student’s GPA varies jointly with the average amount of time he or she studies each week and the average
amount of time spent sleeping each week and varies inversely with the average amount of time spent on social
media each week. Write an equation that relates these quantities.
Example 4: Force of the Wind on a Window
The force F of the wind on a flat surface positioned at a right angle to the direction of the wind varies jointly with
the area A of the surface and the square of the speed v of the wind. A wind of 20 miles per hour blowing on a
window measuring 3 feet by 4 feet has a force of 120 pounds. What force does a wind of 40 miles per hour exert
on a window measuring 4 feet by 5 feet?
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4.1 Linear Function and Their Properties
Average Rate of Change
Linear Functions: f(x)=mx+b
- The average rate of change (slope) is constant for linear functions.
Example 1: Determine if each chart is linear or not.
Time (x) Population (y)
0
0.09
1
0.12
2
0.16
3
0.22
4
0.29
5
0.39
Building Linear Models From Verbal Descriptions
Example 3
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Example 4
Example 5
The simple cost function is the linear cost function,
. Suppose that a small bike manufacturer has a daily cost of $1800 and each bike costs $90 to make.
a) Write a linear model that expresses the cost C of manufacturing x bikes in a day.
b) What is the cost to make 14 bikes in a day?
c) How many bikes can be made for $3780?
Example 6
Suppose the quantity supplied, S, and the quantity demanded, D, of a cell phone each month are given.
S(p)=60p - 900
D(p)= -15p+2850
Where price p is the price in $.
a) Find the equilibrium price (break-even) and quantity
b) Determine the prices for which quantity supplied is greater than quantity demanded.
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4.3 Quadratic Functions and Their Properties
Quadratic Function:
Use the Vertex, Axis of Symmetry, and Intercepts to Graph a Quadratic Function.
Example 1: Locate the vertex and axis of symmetry of the parabola defined by
Example 2: For the quadratic
, answer the following.
a) Does the graph open up or down?
b) What are the coordinates of the vertex?
c) What is the equation of the axis of symmetry?
d) What is/are the x-intercept(s)?
e) What is the y-intercept?
f) Graph the function.
g) Determine the domain and range of the function.
h) Determine where the function is increasing and decreasing.
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Does it open up or down?
Find a Quadratic Function Given Its Vertex and a Point.
1. Plug in vertex as h and k (h, k) and plug in the given point x and y (x, y) to obtain a.
2. Plug a and the vertex in, then simplify.
Example 3: Determine the quadratic function whose vertex is (1, -5) and whose y-intercept is -3.
Example 4: Find the minimum or maximum of the given quadratic.
Example 5
Determine the quadratic function whose vertex is (-2, -16) and whose x-intercepts is 2.
*show MML graphing example
Example 6
A water balloon is launched from a stadium 100 feet above the ground at an inclination of 45° to the horizontal, with an initial velocity of 100 feet per second. From physics, the height
h of the projectile above the ground can be modeled by
where x is the horizontal distance of the projectile from the base of the stadium.
(a) Find the maximum height of the projectile.
(b) How far from the base of the stadium will the water balloon hit the ground?
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4.4 Building Quadratic Models from Verbal description
In economics, revenue R, in dollars, is defined as the amount of money received from the sale of an item and is equal to
the unit selling price p, in dollars, of the item times the number x of units actually sold.
That is, R = xp
Example 1: Maximizing Revenue
An electronics company has found that when certain calculators are sold at a price of p dollars per unit, the number x of
calculators sold is given by the demand equation x = 18,750 – 125p.
(a) Find a model that expresses the revenue R as a function of the price p.
(b) What is the domain of R?
(c) What unit price should be used to maximize revenue?
(d) If this price is charged, what is the maximum revenue?
(e) How many units are sold at this price?
(f) Graph R.
(g) What price should the company charge for the product to collect at least $590,625 in revenue?
Example 2: Maximizing the Area Enclosed by a Fence
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Example 2: Maximizing the Area Enclosed by a Fence
A farmer has 2500 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses
the most area?
Example 3: The Golden Gate Bridge
The Golden Gate Bridge, a suspension bridge, spans the entrance to San Francisco Bay. Its 746-foot-tall towers are 4200
feet apart. The bridge is suspended from two huge cables more than 3 feet in diameter; the 90-foot-wide roadway is
220 feet above the water. The cables are parabolic in shape and touch the road surface at the center of the bridge.
Find the height of the cable above the road at a distance of 1500 feet from the center.
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3.1 Functions
Relation: A relation is a correspondence between two sets: a set X, called the domain, and a set Y, called the range. In a re lation, each
element from the domain corresponds to at least one element from the range.
If x is an element of the domain and y is an element of the range, and if a relation exists from x to y, then we say that y c orresponds to x
or that y depends on x and we write x → y
It is often helpful to think of x as the input and y as the output of the relation.
•
• A relation can be expressed verbally.
• A relation can be expressed numerically using a table of numbers or by using a set of
ordered pairs.
• A relation can be expressed graphically by plotting the points (x, y).
• A relation can be represented as a mapping by drawing an arrow from an element in the
domain to the corresponding element in the range.
A relation is considered a function when no x-values repeat.
Example 1: Describing a Relation
A verbal description of a relation is given below.
The average retail price of gasoline in the U.S. has changed over the years. In 2010, the average cost of one gallon of gasol ine was
$2.78, in 2012 it cost $3.62, in 2014 it cost $3.36, in 2016 it cost $2.14, and in 2018 it cost $2.72.
Using year as input and price as output,
a) What is the domain and the range of the relation?
b) Express the relation as a set of ordered pairs.
c) Express the relation as a mapping.
d) Express the relation as a graph.
Example 2: Determining Whether a Relation Given by a Mapping is a Function
For each relation, state the domain and range. Then determine whether the relation is a function.
a)
b)
Example 3: Determining Whether a Relation Given by a Set of Ordered Pairs Is a Function
For each relation, state the domain and range. Then determine whether the relation is a function.
a) {(1, 5), (3, 9), (5, 1), (9, 2)}
b) {(2, 6), (5, 5), (7, 5), (1, 15)}
c) {(–5, 1), (–2, 2), (3, 3), (2, –2), (–5, –3)}
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Example 4: Determining Whether an Equation Is a Function
Determine whether the equation y = 3x + 7 defines y as a function of x.
Example 5:
Determine whether the equation
defines y as a function of x.
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Use Function Notation; Find the Value of a Function
If f is a function, then for each number x in the domain, the corresponding number y in the range is designated by the symbol f(x), read
as “f of x ” and we write y f x
When a function is expressed in this way, we are using function notation.
We refer to f (x) as the value of the function f at the number x.
For example, the function y = 2x – 5 may be written using function notation as y = f(x) = 2x – 5.
Example 6: Finding Values of a Function
Important Facts about Functions
• For each x in the domain of a function f, there is exactly one image f(x) in the range; however, more than one x in the domain can have
the same image in the range.
• f is the symbol that we use to denote the function. It is symbolic of the equation (rule) that we use to get from x in the domain to
f(x) in the range.
• If y = f(x), then x is the independent variable, or the argument of f, and y, or f(x), is the dependent variable, or the value of f at x.
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Difference Quotient
Example 7: Finding the Difference Quotient
Find the difference quotient of each function.
a)
b)
Finding the Domain of a Function Defined by an Equation
• If there is no fraction with a variable in the denominator or radical the domain is all real numbers.
• If the equation has a fraction with a variable in the denominator, set the denominator equal to zero and solve. We do this b ecause we
cannot have 0 in the denominator.
• If the equation is just a radical or a fraction with a radical in the denominator, set the radicand (inside) greater than or equal to zero.
We do this because we cannot have negative numbers inside the radical.
• If you have an equation with a fraction having a radical on top and a variable in the denominator, set the radicand greater t han or
equal to zero AND set the denominator equal to zero.
Example 8: Find the Domain of a Function
Example 2: Find the domain of each of the following:
a)
b)
c)
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d)
Form the Sum, Difference, Product, and Quotient of Two Functions
Example 9: Operations on Function
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3.2 The graph of a Function
Use the vertical line test.
Example 1: Which graphs are graphs of a function?
Obtain Information From a Graph.
Example 2:
d) Is f(-35) positive or
negative?
g) What is the range of f?
e) For what value(s) of
x is f(x)=0
h) What are the x-intercepts?
i) What are the y-intercepts?
f) For what values of x
is f(x)>0?
j) How often does the line y=1 intersect the
graph?
a) Find F(-35) and f(-15)
b) Find f(30) and f(0)
g) What is the domain
of f?
c) Is f (10) positive or negative?
Example 3
Determine whether the graph is that is of a function by using the vertical line test.
a) What is the domain and range?
b) What are the intercepts?
c) Find any symmetry.
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l) For what value(s) of x does f(x)=15?
Example 4:
a)
b)
c)
d)
e)
f)
Is the point (-1, 2) on the graph of F?
If x=4, what is f(x)? What point is on the graph of F?
If f(x)=2, what is x? What point(s) are on the graph of f?
What is the domain of f?
List the x-intercepts, if any?
List the y-intercepts, if any?
Example 5
A golf ball is hit with an initial velocity of 130ft/s at an inclination of 45 degrees to the horizontal. In physics, it is established that the height h
of the golf ball is given by the function
where x is the horizontal distance that the golf ball has traveled.
a) Determine the height of the golf ball after it has traveled 100ft.
b) What is the height after it has traveled 300ft?
c) How far is the golf ball hit when it lands?
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3.3 Properties of Functions
Determine Even and Odd Functions from a Graph
Even: y-axis symmetry
Odd: origin symmetry
Example 1
Example 2: Identifying Even and Odd Functions from a Graph
Determine whether each graph is the graph of an even function, an odd
function, or a function that is neither even nor odd.
Is f strictly decreasing on the interval (-2, 1)?
Identify Even and Odd Functions from the Equation
Example 2
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Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant
Example 3
Determine where the function is increasing, decreasing, or constant. List the intercepts, if any, and find the domain and ra nge.
Use a Graph to Locate Local Maxima and Local Minima
Example 4
Use a Graph to Locate the Absolute Maximum and the Absolute Minimum
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Find the Average Rate of Change
Example 5
Find the average rate of change of
a) From 1 to 3 b) From 2 to 4
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3.4 Library of Functions; Piecewise-Defined Functions
Library of Functions
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Piecewise-Defined Functions
Example 1
Example 2
Graph the piece-wise function:
If
a)
b)
c)
d)
Find the domain of the function.
Locate and intercepts.
Based on the graph, find the range.
Is F continuous on its domain?
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Example 3
Graph, find the intercepts, domain, range.
Example 4
A telephone company offers a monthly phone plan of $29.99. It includes 250 anytime minutes plus $0.30 per minute for additional minutes. The following
function is used to compute the monthly cost for a subscriber, where x is the number of anytime minutes used.
Compute the monthly cost of the cellular phone for the use of:
a) 130min
b) 315min
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3.5 Graphing Techniques: Transformations
Example 1: Vertical Shift Up
Use the graph of
to obtain the graph of
Find the domain and range of g.
Example 2: Vertical Shift Down
Use the graph of
to obtain the graph of
Find the domain and range of g.
Vertical Shifts
If a positive real number k is added to the output of a function y = f(x), the graph of the new
function y = f(x) + k is the graph of f shifted vertically up k units.
If a positive real number k is subtracted from the output of a function y = f(x), the graph of
the new function y = f(x) – k is the graph of f shifted vertically down k units.
Example 3: Horizontal Shift to the Right
Use the graph of f(x) = |x| to obtain the graph of g(x) = |x – 2|.
Find the domain and range of g.
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Example 4: Horizontal Shift to the Left
Use the graph of f(x) = |x| to obtain the graph of g(x) = |x + 2|.
Find the domain and range of g.
Horizontal Shifts
If the argument x of a function f is replaced by x – h, h > 0, the graph of the new function
y = f(x – h) is the graph f shifted horizontally right h units.
If the argument x of a function f is replaced by x + h, h > 0, the graph of the new function
y = f(x + h) is the graph f shifted horizontally left h units.
Example 5: Combining Vertical and Horizontal Shifts
Graph the function
. Find the domain and range of f.
Example 6: Vertical Stretch
Use the graph of
to obtain the graph of
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Example 7: Vertical Compression
Use the graph of to obtain the graph of
of
.
Vertical Compression or Stretch
If a function y = f(x) is multiplied by a positive number a, then the graph of the new function
y = af(x) is obtained by multiplying each y-coordinate of the graph of y = f(x) by a.
If 0 < a < 1, a vertical compression by a factor of a results.
If a > 1, a vertical stretch by a factor of a results.
Horizontal Compression or Stretch
If the argument of a function y = f(x) is multiplied by a positive number a, then the graph of
the new function y = f(ax) is obtained by multiplying each
x-coordinate of the graph of y = f(x) by
If a > 1, a horizontal compression by a factor of
If 0 < a < 1, a horizontal stretch by a factor of
results.
results.
Example 8: Reflection about the x-Axis
Graph the function
. Find the domain and range of f.
Reflection about the x-Axis
When a function f is multiplied by –1, the graph of the new function y = –f(x) is the reflection
about the x-axis of the graph of the function f.
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Example 9: Reflection about the y-Axis
Graph the function
.Find the domain and range of f.
Reflection about the y-Axis
When the graph of the function f is known, the graph of the new function y = f(–x) is the
reflection about the y-axis of the graph of the function f.
Example 10
Find the function that is finally graphed after the following transformation are applied to a
graph of y
x in the order listed.
(1) Shift up 10 units
(2) Reflect about the x-axis
(3) Reflect about the y-axis
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3.6 Mathematical Models: Building Functions
Build and Analyze Functions
Example 1:
Example 2:
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Example 3:
A rectangle has a corner in Quadrant 1 on the graph of
the postive x-axis.
a) Draw the figure.
b) Express the area A of the rectangle as a function of x.
c) What is the domain of A?
1105 11e Notes Page 46
another at the orgin, a third on the postive y-axis, and fourth on
6.1 Composite Functions
Find a Composite Function and Domain
To form a composite:
- Put the second letter (that function) into the first letter (function) for every x and solve.
Domain of a composite:
- Find domain of the function you inputted (second letter).
- Find the domain of the composite.
Example 1
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Example 2
For
a)
b)
Example 3
Find the composite functions and state the domain.
a)
b)
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6.2 One-to-One Functions; Inverse Functions
Determine Whether a Function Is One-to-One
A one-to-one function: Each x in the domain has one and only one image in the range. Meaning one x to one y and vice versa.
Example 1: Determine which figure represents a one-to-one function
Example 2
Determine the Inverse of a Function Defined by a Map or a Set of Ordered Pairs
Example 3
1105 11e Notes Page 49
Example 4
Consider the functions
a) Find f(g(x)).
b) Find g(f(x)).
c) Determine whether the functions f and g are inverses of each other.
Find the Inverse of a Function Defined by an Equation
Meaning: Switch x and y and solve for y.
Example 5
Find the inverse if
and check your answer.
b) Find the domain and the range of f and
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Example 6
Example 7
The graph of a one-to-one function is given. Draw the graph of the inverse function.
Example 8
The domain if a one-to-one function is [5, ∞), and its range is [-2, ∞), State the domain and range of its inverse function.
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6.3 Exponential Functions
Malthusian catastrophe as predicted by Malthus states that the growth of human population is governed by the rate of crop production. In the future, if the growth rate
of crops decreases then human population growth is decreased. And there might come a day when we may have extremely limited supply of food which can cause
widespread hunger and starvation.
So exponential function is a reminder of how efficiently do we need utilize the available resources.
"There is a continuous cycle of innovation that is necessary in order to sustain and avoid collapse. The catch, however, to this is that you have to innovate faster and
faster. So the image is that we're not only on a treadmill that's going faster and faster, but we have to change the treadmill faster and faster. We have to accelerate on a
continuous basis." - Geoffrey West
Example 1
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Graph Exponential Functions
Example 2
Example 3: Demo of MML
Graphing Exponential Functions Using Transformations.
Solve Exponential Equations
Example 4
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Plot the point (0, 1), then move onto
shifting and changing the base.
Number e
e
It is approximately equal to
2.71828
Example 5
Solve:
Example 6
Solve
Example 7: Exponential Probability
Between 9:00 pm and 10:00 pm, cars arrive at In-N-Out's drive-thru at the rate of 15 cars per hour (0.25 car per minute). The following formula from statistics can be
used to determine the probability that a car will arrive within t minutes of 9:00 pm.
F(t) = 1 – e–0.25t
a) Determine the probability that a car will arrive within 5 minutes of 9 pm (that is, before 9:05 pm).
b) Determine the probability that a car will arrive within 30 minutes of 9 pm (before 9:30 pm).
c) Graph F.
d) What does F approach as t increases without bound in the positive direction?
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6.4 Logarithmic Functions
Human senses, nearly all, work in a manner and obey Weber–Fetcher law, that response of the sense machinery is logarithm of an input. It is
true at least for hearing, but also for eye sensitivity, temperature sense etc. And of course, in areas where it works normal ly. Because in
extreme, there are other processes such as pain, etc.
So as in a cause of hearing, what you experience is the logarithm of power of a sound wave, by "biological, natural, hear sen se construction.
So, it is natural to use logarithmic units.
Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements
Example 1
Example 2
Example 3
Use a calculator to evaluate the expression. Round to three decimal places.
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Evaluating Logarithmic Functions
Example 4
Determine the Domain of a Logarithmic Function
The domain of a logarithmic function consists of the positive real numbers, so the argument of a logarithmic function must be greater than
zero.
Example 5
Solve Logarithmic Equations
Example 6
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Example 7
Example 8
Solve the equation. Write in terms of the common logarithm.
Example 9: Graphing a Logarithmic Function and Its Inverse
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Example 10: Graphing a Logarithmic Function and Its Inverse
Example 11: Alcohol and Driving
Blood alcohol concentration BAC is a measure of the amount of alcohol in a person’s bloodstream A BAC of
% means that a person has
4 parts alcohol per 10,000 parts blood in the body. Relative risk is defined as the likelihood of one event occurring divided by the likelihood
of a second event occurring. For example, if an individual with a BAC of 0.02% is 1.4 times as likely to have a car accident as an individual
who has not been drinking, the relative risk of an accident with a BAC of 0.02% is 1.4.
a) Research indicates that the relative risk of a person having an accident with a BAC of 0.02% is 1.4. Find the constant k i n the equation.
b) Using this value of k, what is the relative risk if the concentration is 0.17%?
c) Using this same value of k, what BAC corresponds to a relative risk of 100?
d) If the law asserts that anyone with a relative risk of 4 or more should not have driving privileges, at what concentration of alcohol in the
bloodstream should a driver be arrested and charged with DUI (driving under the influence)?
1105 11e Notes Page 62
6.5 Properties of Logarithms
Work With Properties of Logarithms
Example 1
Write each expression as a sum and/or difference of logarithms. Express powers as factors.
a) log
b) log
c) ln
d) ln
e) ln
1105 11e Notes Page 63
Example 2
Write each of the expressions as a single logarithm.
a) log
log
b) log
log
c)
log
log
log
Evaluate Logarithms Whose Base Is Neither 10 Nor e
Example 3
Use the change-of-base formula and a calculator to evaluate the logarithm
a) log
b) log
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6.6 Logarithmic and Exponential Equations
Example 1
Example 2
Example 3
Solve Exponential Equations
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Solve Exponential Equations
Example 4
Example 5
Solve: a)
b)
Example 6
1105 11e Notes Page 66
Example 6
Solve:
Example 7: Solving an Exponential Equation That Is Quadratic in Form
Solve:
1105 11e Notes Page 67
6.7 Financial Models
Continuous Compounding Theorem
The amount A after t years due to principal P invested at an annual interest rate r compounded continuously is
A = Pert
Example 1: Using Continuous Compounding
The amount A that results from investing a principal P of $2000 at an annual rate r of 8% compounded
continuously for a time t of 1 year is:
Effective Rate of Interest Theorem
Example 2: Computing the Effective Rate of Interest-Which Is the Best Deal?
Suppose you want to buy a 5-year certificate of deposit (CD). You visit three banks to determine their CD rates.
American Express offers you 2.17% annual interest compounded monthly, and First Internet Bank offers you
2.18% compounded quarterly. Discover offers 2.16% compounded daily. Determine which bank is offering the
best deal.
The bank that offers the best deal is the one with the highest effective interest rate.
Present Value Formulas
1105 11e Notes Page 68
Present Value Formulas
Example 3: Computing the Value of a Zero-Coupon Bond
A zero-coupon (noninterest-bearing) bond can be redeemed in 10 years for $1000. How much should you be
willing to pay for it now if you want a return of 6% compounded continuously?
Example 4: Determining the Time Required to Double or Triple an Investment
a) How long will it take for an investment to double in value if it earns 6% compounded continuously?
b) How long will it take to triple at this rate?
1105 11e Notes Page 69
Exponential Growth and Decay Models; Newton’s Law
Law of Uninhibited Growth or Decay
Many natural phenomena follow the law that an amount A varies with time t according to the function
A(t) = A0ekt
Here A is the original amount (t = 0) and k ≠ is a constant If k > 0, then equation (1) states that the
amount A is increasing over time; if k < 0, the amount A is decreasing over time. In either case, when an
amount A varies over time according to equation (1), it is said to follow the exponential law, or the law
of uninhibited growth (k > 0) or decay (k < 0).
Uninhibited Growth of Cells
A model that gives the number N of cells in a culture after a time t has passed (in the early stages of
growth) is N(t) = N0ekt k > 0, where N0 is the initial number of cells and k is a positive constant that
represents the growth rate of the cells.
Example 1: Bacterial Growth
A colony of bacteria that grows according to the law of uninhibited growth is modeled by the function
N(t) = 90e0.06t, where N is measured in grams and t is measured in days.
a) Determine the initial amount of bacteria.
b) What is the growth rate of the bacteria?
c) What is the population after 7 days?
d) How long will it take for the population to reach 250 grams?
e) What is the doubling time for the population?
Example 2: Bacterial Growth
1105 11e Notes Page 70
Example 2: Bacterial Growth
A colony of bacteria increases according to the law of uninhibited growth.
a) If N is the number of cells and t is the time in hours, express N as a function of t.
b) If the number of bacteria doubles in 5 hours, find the function that gives the number of cells in the
culture.
c) How long will it take for the size of the colony to triple?
d) How long will it take for the population to double a second time (that is, increase four times)?
Uninhibited Radioactive Decay
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Uninhibited Radioactive Decay
The amount A of a radioactive material present at time t is given by A(t) = A0ekt k < 0, where A0 is the
original amount of radioactive material and k is a negative number that represents the rate of decay.
Example 3: Estimating the Age of Ancient Tools
Traces of burned wood along with ancient stone tools in an archeological dig were found to contain
approximately 2.33% of the original amount of carbon-14. If the half-life of carbon-14 is 5730 years,
approximately when was the tree cut and burned?
Newton’s Law of Cooling
The temperature u of a heated object at a given time t can be modeled by the following function:
u(t) = T + (u0 – T)ekt k < 0 , where T is the constant temperature of the surrounding medium, u0 is the initial
temperature of the heated object, and k is a negative constant.
Example : Using Newton’s Law of Cooling
An object is heated to 90°C (degrees Celsius) and is then allowed to cool in a room whose air
temperature is 20°C.
a) If the temperature of the object is 75°C after 5 minutes, when will its temperature be 50°C?
b) Determine the elapsed time before the temperature of the object is 35°C.
c) What do you notice about the temperature as time passes?
1105 11e Notes Page 72
5.1 Polynomial Functions
Polynomial Function
A polynomial function in one variable is a function of the form
(standard form (highest exponent to lowest exponent))
where a , a
… a , a are constants, called the coefficients of the polynomial, n ≥ 0 is an integer, and x is a variable. If
a ≠ 0, it is called the leading coefficient, and n is the degree of the polynomial.
The domain of a polynomial function is the set of all real numbers.
Example 1: Identifying Polynomial Functions
Determine which of the following are polynomial functions. For those that are, state the degree; for those that are not,
state why not. Write each polynomial function in standard form, and then identify the leading term and the constant
term.
Summary of Properties
If you take a course in calculus, you will learn that the graph of every polynomial function is both smooth and
continuous. By smooth, we mean that the graph contains no sharp corners or cusps; by continuous, we mean that the
graph has no gaps or holes and can be drawn without lifting your pencil from the paper.
•
•
•
•
Real Zero
If f is a function and r is a real number for which f(r) = 0, then r
is called a real zero of f.
As a consequence of this definition, the following statements are
equivalent.
r is a real zero of a polynomial function f.
r is an x-intercept of the graph of f.
x – r is a factor of f.
r is a real solution to the equation f(x) = 0.
1105 11e Notes Page 73
•
•
•
•
r is a real zero of a polynomial function f.
r is an x-intercept of the graph of f.
x – r is a factor of f.
r is a real solution to the equation f(x) = 0.
Identifying the Real Zeros of a Polynomial Function and Their Multiplicity
If a polynomial function f is factored completely, it is easy to locate the x-intercepts of the graph by solving the equation
f(x) = 0 using the Zero-Product Property (setting the equation equal to zero).
Example 2: Finding a Polynomial Function from Its Real Zeros
Find a polynomial function of degree 3 whose real zeros are –1, 2, and 3.
Example 3: Identifying Real Zeros and Their Multiplicities
1105 11e Notes Page 74
Example 4: Graphing a Polynomial Function Using Its x-intercept
Consider the polynomial function:
a) Find the x- and y-intercepts of the graph of f.
b) Use the x-intercepts to find the intervals on which the graph of f is above the x-axis and the intervals on which the
graph of f is below the x-axis.
c) Locate other points on the graph, and connect the points with a smooth, continuous curve.
Interval
x
y
A/B?
Turning Points
If f is a polynomial function of degree n, then the graph of f has at most n – 1 turning points.
(If given turning points the degree of the polynomial can be found by adding 1 to the number of turning points on the
graph)
Example 5: Identifying the Graph of a Polynomial Function
Which graphs in the figures could be the graph of a polynomial function? For those that could, list the real zeros and
state the least degree the polynomial can have. For those that could not, say why not.
1105 11e Notes Page 75
End Behavior of the Graph of a Polynomial Function
The end behavior of the graph of the polynomial function
is the same as that of the graph of the power function
(y = LT)
Example 6: Identifying the Graph of a Polynomial Function
Example6: Which graph in the figures could be the graph of
f (x) = x5 + ax3 + bx2 – 2x + 3 where a > 0, b > 0?
Example 7: Finding a Polynomial Function from a Graph
Find a polynomial function whose graph is shown (use the smallest degree possible).
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Find a polynomial function whose graph is shown (use the smallest degree possible).
Example 8: Find the polynomial
Given: Zeros: -2, multiplicity 2; 0; 5, multiplicity 2; Degree: 5
1105 11e Notes Page 77
5.2 Graphing Polynomial Functions; Models
Example 1: Graphing a Polynomial Function
Graph the polynomial function f(x) = (3x + 1)(x – 2)2.
Example 2: Graphing a Polynomial Function
Graph the polynomial function
f(x) = (x + 1)(x + 3)2(x – 4).
1105 11e Notes Page 78
Example 3: Graphing a Polynomial Function Using a Graphing Utility
Analyze the graph of the polynomial function:
f(x) = x3 + 2.84x2 – 4.2251x + 2.34856
The figures shown are from a TI-84 Plus C.
The y-intercept is f(0) = 2.34856.
The graph of f
Since it is not readily apparent how to factor f, use a
graphing utility's ZERO (or ROOT or SOLVE) feature
and determine the x-intercept is –4.03, rounded to two
decimal places.
The table shows values of x on each side of the x-intercept using a TI-84 Plus C.
The points (–5, –30.53) and (–3, 13.58) rounded to two decimal places, are on the graph.
From the graph of f, we see that f has two turning points.
Using MAXIMUM reveals one turning point is at (–2.46, 15.04), rounded to two decimal places.
1105 11e Notes Page 79
Using MINIMUM shows that theBased
other turning
point fisisatincreasing
(0.57, 1.05),
rounded to two decimal
on the graph,
on the
places.
intervals (–∞ –
and
∞
Also, f is decreasing on interval [–2.46, 0.57].
The figure shows the graph of f drawn by hand using the information in the previous steps
1105 11e Notes Page 80
5.3 Properties of Rational Functions
Rational Function
A rational function is a function of the form
where p and q are polynomial functions and q is not the zero polynomial.
The domain of R is the set of all real numbers, except those for which the denominator q is 0.
Meaning: A rational function is when you have a fraction that has a polynomial on top and a polynomial
on bottom.
The domain is all real numbers, except the number that makes the bottome zero. Set
the denominator equal to zero and solve. Exclude that value.
Example 1: Finding the Domain of a Rational Function
Horizontal and Vertical Asymptotes
Let R denote a function.
- If, as x → –∞ or as x → ∞ the values of R(x) approach some fixed number L, then the line y = L is a
horizontal asymptote of the graph of R.
- If, as x approaches some number c, the values |R(x → ∞ that is R(x → –∞ or R(x → ∞ then the
line x = c is a vertical asymptote of the graph of R.
Locating Vertical Asymptotes (VA)
The graph of a rational function
,
1105 11e Notes Page 81
The graph of a rational function
,
in lowest terms, has a vertical asymptote x = r if r is a real zero of the denominator q. That is, if x – r is a
factor of the denominator q of the rational function R, in lowest terms, the graph of R has a vertical
asymptote x = r.
Meaning: The vertical asymptote is the domain.
Example 2: Finding Vertical Asymptotes
Finding Horizontal or an Oblique Asymptote of a Rational Function
Example 3
1105 11e Notes Page 82
Example 3
Example 4
Example 5
Find the horizontal or oblique asymptote, if one exists, of the graph of:
Example 6
Find the horizontal or oblique asymptote, if one exists, of the graph of:
1105 11e Notes Page 83
Find the horizontal or oblique asymptote, if one exists, of the graph of:
Example 6
Example 7
1105 11e Notes Page 84
8.1/8.6/8.7
8.1 Systems of Equations
2 variable systems
Solve Systems of Three Equations Containing Three Variables
Example 1
Solve the system. If the system has no solution, say it is inconsistent.
Example 2
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Example 3
Example 4
1105 11e Notes Page 86
8.6 Systems of Nonlinear Equations
Example 1
Solve the system.
Example 2
1105 11e Notes Page 87
Example 3
8.7 Systems of Inequalities
Example 1
Graph the inequality
1105 11e Notes Page 88
Example 2
Graph
*show graphing demo
Example 3
A retired couple can invest up to $25,000. As their financial adviser, you recommend that they place at least $15,000 in Treasury bills
yielding 2% and at most $5000 in corporate bonds yielding 3%.
a) Using x to denote the amount of money invested in Treasury bills and y to denote the amount invested in corporate bonds, write
a system of linear inequalities that describe the possible amounts of each investment. Assume that x and y are in thousands of
dollars.
b) Graph the system
c) What are the corner points of the graph?
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1105 11e Notes Page 90