Unit 1: Basic Functions and Rational Functions
Definitions, Properties & Formulas
Linear Equation
Slope
equation of a straight line
the slope, m, of the line through (x1, y1) and (x2, y2) is given by the following
y  y1
equation, if x1  x2: m  2
x 2  x1
y
Types of Slope
Positive
y
x
Negative
x
y-intercept
where the graph crosses the y-axis
x-intercept
where the graph crosses the x-axis
Slope-Intercept
Form
Standard Form
Point-Slope
Form
Parallel Lines
Perpendicular
Lines
y
y
x
Zero
horizontal line:
y=b
x
Undefined
vertical line: x
=a
y = mx + b
where m represents the slope and b represents the y-intercept of the linear
equation
Ax + By = C
where A, B, and C are constants and A  0 (positive, whole number)
y – y1 = m(x – x1)
where m represents the slope and (x1, y1) are the coordinates of a point on the line
of the linear equation
Two nonvertical lines in a plane are parallel if and only if their slopes are equal
and they have no points in common. (Two vertical lines are always parallel.)
ex)
y = 2x + 3
m=2
and
y = 2x – 4
m=2
 equal slopes
 // lines
Two nonvertical lines in a plane are perpendicular if and only if their slopes are
negative reciprocals. (A horizontal and a vertical line are always perpendicular.)
5
2
y  x 7
y   x 1
ex)
and
 neg. recip. slopes
2
5
5
2
  lines
m=
m= 
5
2
Relation
a set of ordered pairs (x, y)
Domain
the set of all x-values of the ordered pairs
Range
the set of all y-values of the ordered pairs
1
Function
Vertical Line
Test (VLT)
Horizontal Line
Test (HLT)
One-to-One
Functions
Inverse Relations
& Functions
Writing Inverse
Functions
a relation in which each element of the domain is paired with exactly one element
in the range.
If any vertical line passes through two or more points on the graph of a relation,
then it does not define a function.
If any horizontal line passes through two or more points on the graph of a
relation, then its inverse does not define a function.
a function where each range element has a unique domain element
(use HLT to determine)
f -1(x) is the inverse of f(x), but f -1(x) may not be a function
(use HLT to determine)
To find f -1(x):
(1) let f(x) = y
(2) switch the x and y variables
(3) solve for y
(4) let y = f -1(x)
Odd Functions
symmetric with respect to the origin  TEST: f(-x) = -f(x)
Even Functions
symmetric with respect to the y-axis  TEST: f(-x) = f(x)
Symmetry Tests
Operations with
Functions
symmetric with respect to the:
y-axis
x-axis
origin
the given equation is equivalent when:
x is replaced with -x
y is replaced with –y
x and y are replaced with –x and -y
sum:
(f + g)(x) = f(x) + g(x)
difference:
(f – g)(x) = f(x) – g(x)
product:
(f  g)(x) = f(x)  g(x)
quotient:
f
f ( x)
 ( x ) 
, where g( x )  0
g( x )
 g
2
Composition of
Functions
Zeroes
given functions f and g, the composite function is f  g( x)  f g( x) , where g(x)
is substituted for x in the f(x) function
set numerator equal to zero and solve
Vertical
Asymptotes
set denominator equal to zero and solve for x (beware of the “hole” !)
Horizontal
Asymptotes
Case:
Horizontal
Asymptote
Oblique
Asymptotes
use long division to find the quotient of the numerator and denominator. Only
works if m = n + 1
Point
Discontinuities
Factor the numerator and the denominator.
Cancel the "like" factors (if any)
Off on the side, set the cancelled factor equal to 0 and solve. This is the xvalue of the hole in the function.
Plug the "x-value" into the simplified function and solve for y. This is the yvalue of the hole in the function.
Graph the simplified function by finding zeroes, VAs, and HAs. Plot the
location of the hole.
Practical
Domain
a domain that makes sense within the context of the given problem (generally
we avoid negative measures)
mn
y0
mn
y
am
an
mn
none
3
Precalculus
Review- Test- Basic Functions and Rational Functions
Name:______________________________
Date:_______________________________
(1)
Which of the following equations define functions? Explain your reasoning.
(a) y = x + 6
(b) y2 = x + 1
(c) y3 = x + 4
(d) y = x – 2
(e) y3 = x – 3
(f) y2 = x – 5
(2)
Which of the following functions are one-to-one? Explain your reasoning.
(a) f(x) = x5
(b) g(x) = 2x + 7
(c) h(x) = x2 – 4
(3)
Write the linear equation in standard form Ax + By = C that passes thru the points (3, 5) & (4, -3).
(4)
Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3)
and is parallel to the line 2x + 2y = 5.
(5)
Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3)
and is perpendicular to the line 3x – 3y = 4.
(6)
Write the linear equation in standard form Ax + By = C that is a horizontal line and passes
through the point (-9, 2).
(7)
Given f(x) = 2x2 – x + 3 find f(k – 2)
(8) Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1)
(9)
For the function f(x) = -6x + 5, find and simplify:
f ( x  h)  f ( x )
, h0
(a)
h
(b)
f ( x )  f (a )
, x a  0
x a
4
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(10)
Find the domain of each of the following:
2x  3
(a) f ( x ) 
2x
(b) g( x )  x  4
__________________________________________________________________________________________
(11)
Algebraically determine the symmetry with respect to the y-axis, x-axis, and origin, if any exists, for
each of the given equations:
2x  4 y  7
a)
b)
9 x 2  4 y 2  36
__________________________________________________________________________________________
(12) Given f ( x)  x 2  3 and g ( x)  2 x  4 , find  f  g (x) and g  f (x) and give each domain.
__________________________________________________________________________________________
For each of the following functions, f (x) , find the inverse, f 1 ( x) :
f ( x)  5 x  2
a)
4
f ( x) 
b)
x3
__________________________________________________________________________________________
y
(14) Given the graph on the right, answer the following questions:
(13)
(a)
write the linear equation in slope-intercept form:
(b)
write in slope-intercept form the equation of the parallel line
through (2, -1) and graph:
(c)
x
write in slope-intercept form the equation of the
perpendicular line through the x-intercept and graph:
__________________________________________________________________________________________
(15)
Perform the four basic operations with the functions f(x) = 4x and g(x) = x2 + 2
(f + g)(x) =
(f – g)(x) =
(f  g)(x) =
f
 ( x ) 
 g
__________________________________________________________________________________________
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(16)
The Great Stride Shoe Company determines that the annual cost of making x pairs of one type of shoe is
$30 per pair plus $100,000 in fixed overhead costs. Each pair of shoes that is manufactured is sold
wholesale for $50.
(a)
Write a linear function C(x) that models the cost of producing x pairs of shoes.
(b)
Write a linear function R(x) that models the revenue produced from selling x pairs of shoes.
(c)
Find how many pairs of shoes must be made and sold in order to break even.
(d)
Sketch the graphs from parts (a) and (b) and indicate the WINDOW used.
(e)
Explain the graphical interpretation of your answer
to part (c).
__________________________________________________________________________________________
(17)
Bermuda grass can be found in Africa and Asia, and has an amazing growth rate of 5.9 inches per
day. Suppose you cut a Bermuda grass plant to a length of 2 inches.
(a)
Write a linear function L(x) that models the length of the plant after x days.
(b)
Using your function from (a), find the length of the plant if you didn’t cut it again for one week.
__________________________________________________________________________________________
(18)
A digital camera was purchased for $250 and is assumed to have a salvage value of $40 after 7 years.
Its value has depreciated linearly during this period.
(a)
Write a linear function V(t) that relates the value of the camera in dollars to time t in years.
(b)
What would be the depreciated value of the camera after 2 years? 5 years?
(c)
What is the practical domain of the depreciation function?
(d)
Sketch the graph and indicate the WINDOW used.
__________________________________________________________________________________________
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(19)
Identify the domain, zeroes, asymptotes, holes, and graph the function:
__________________________________________________________________________________________
(20)
Identify the domain, zeroes, asymptotes, holes, and graph the function:
__________________________________________________________________________________________
(21)
Identify the domain, zeroes, asymptotes, holes, and graph the function:
__________________________________________________________________________________________
(22)
Identify the domain, zeroes, asymptotes, holes, and graph the function:
__________________________________________________________________________________________
(23)
Identify the domain, zeroes, asymptotes, holes, and graph the function:
__________________________________________________________________________________________
(24)
A rectangular area of 100 square meters has a wall
on one of its sides, as shown. The sides
perpendicular to the wall are made of fencing that
costs $40 per meter. The side parallel to the wall is
made of decorative fencing that costs $50 per meter.
a.
Write a function to express the total cost, C(x), of the
fencing as a function of x.
b.
Find the minimum cost to the nearest dollar. What are
the dimensions that give this minimum cost to the
nearest tenth?
c.
Sketch the graph and indicate the WINDOW used.
x
L
L
x
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