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0.5 A Basic Library of Functions
Algebraic functions: A function is considered algebraic if it can be expressed in terms of constants and
the variable x by using only arithmetic operations (addition, subtraction, multiplication, and division)
and rational constant powers of the variable.
Example)
Non-example)
Graphing Algebraic Functions
Domain: all the things x can be
Things to consider when determining the domain:
a. The domain is usually all reals.
b. The domain is restricted if there is a variable in the denominator.
c. The domain is restricted if there is a variable under the radical.
Range: all the things y can be
Things to consider when determining the range. What happens if…
a.
b.
c.
d.
e.
f.
x=0
x is positive versus negative
x is in the denominator
the function contains a radical
the function contains an absolute value
the function contains a fraction with a variable
If a graph is a function, it will pass the vertical line test.
The key to graphing is recognizing different types of graphs from their equations. On the chart below,
sketch the parent graph and state the domain and range.
1
Type of
function
Equation
linear
f  x   mx  b
quadratic
f  x   ax2  bx  c
Parent graph
or
(polynomial)
cubic
f  x   ( x  h)2  k
f(x) = ax3 + bx2 + cx +d
(polynomial)
quartic
f(x) = ax4 + bx3 + cx2 + dx + e
(polynomial)
power
f(x) = Axk
A is not 0, k is rational
absolute value
f(x) = a x  b + c
2
Domain and
Range
xa b
square root
f(x) =
cube root
f(x) =
rational
f(x) = fraction
3
xa b
*the numerator AND
denominator are
polynomials
Ex.) Classify the given functions as Algebraic, Non-algebraic, Linear, Power, Polynomial, Rational, or any
combination of these functions
f(x) = x-5
h(x) =
x 1
x 1
g(x) = 2x – 1
k(x) = 8x3 + 2x2 – 2x
3
LINEAR FUNCTIONS
Slope is calculated by finding
y f (b)  f (a )

which we learned in the previous section can be
x
ba
thought of as “average rate of change”
Parallel Lines: have the same slope
Perpendicular Lines: have negative reciprocal slopes
All linear functions can be thought of as f(x) = mx + b where m and b are real numbers, m represents the
slope, and b represents the y-intercept.
Properties of Linear Functions:
Suppose f(x) = mx + b is a linear function. Then
(a) the domain of f(x) is
(b) the range of f(x) is
if m  0 and is {b} if m = 0
(c) the average rate of change of f(x) on any interval is m
(d) the graph of f(x) is a line with slope m and y-intercept b
Forms of a Linear Equation
Slope-Intercept Form: Given slope m and y-intercept b,
y = mx + b
Point-Slope Form:
Given slope m and a point (x0, y0),
y – y0 = m(x – x0)
Two-Point Form:
Given two points (x0, y0) and (x1, y1),
y y 
y  y0   1 0  ( x  x0 )
 x1  x0 
Vertical lines:
xa
ex.)
x=4
Horizontal lines:
yb
ex.)
y=6
4
Exercises: Write an equation of a line that satisfies the following conditions: (Answers at the end.)
1. The line passes thru the point (2, 3) and has slope equal to -3/2.
2. The line passes thru the points (-2, -1) and (3, 4).
3. The line passes thru the point (-1, 2) and is parallel to y = 3x – 4.
4. The line passes thru the points (5, 3) and (5, -4).
5. The line has a y-intercept of -5 and passes thru the point (2, -1).
Answers: 1. y  
3
x  6 or 3 x  2 y  12
2
2. y  x  1 3. y  3 x  5
4. x  5 5. y  2 x  5
ABSOLUTE VALUE FUNCTIONS
Absolute Value Functions can be viewed as Piecewise-Linear graphs. Your book defines the
 g ( x), for all x with g ( x)  0
absolute value f of a function g to be f ( x)  g ( x)  
 g ( x), for all x with g ( x)  0
f(x) = |2x – 1|
6
4
2
-10
-5
5
10
-2
-4
-6
5
TRANSCENDENTAL FUNCTIONS
Any function that is not Algebraic is classified as a transcendental function.
Type of
function
Equation
exponential
f(x) = ax + b, a > 0,
Parent graph
a≠1
logarithmic
f(x) = log a x , a > 0,
a≠1
trigonometric:
sine
f(x) = a sin(bx  c)  d
trigonometric:
cosine
f(x) = a cos(bx  c)  d
6
Domain and
Range