Properties of Functions
A function is defined as a set of all ordered pairs (x, y), such that for each element x, there corresponds exactly one element y.
The domain of is the set x.
The range of is the set y.
Combinations of Functions
If (x) = 3x + 1 and g(x) = x2 - 1
a) the sum (x) + g(x) = (3x + 1) + (x2 - 1) = x2 + 3x
b) the difference (x) - g(x) = (3x + 1) - (x2 - 1) = -x2 + 3x + 2
c) the product (x)g(x) = (3x + 1)(x2 - 1) = 3x3 + x2 - 3x - 1
d) the quotient (x)/g(x) = (3x + 1)/(x2 - 1)
e) the composite ( ° g)(x) = (g(x))
= 3(x2 - 1) + 1 = 3x2
Functions and g are inverses of each other if
(g(x)) = x for each x in the domain of g
g((x)) = x for each x in the domain of
The inverse of the function is denoted -1.
To find -1, switch x and y
in the original equation and solve the equation for y in
terms of x.
|Exercise:||If (x) = 3x + 2, then -1(x) =|
|(B) - 2|
|(C) 3x - 2|
|(D) x + 3|
|The answer is E.||x = 3y + 2|
|3y = x - 2|
Even and Odd Functions
The function y = (x) is even if (-x) = (x).
Even functions are symmetric about the y-axis (e.g. y = x2)
The function y = (x) is odd if (-x) = -(x).
Odd functions are symmetric about the origin (e.g. y =
|Exercise:||If the graph of y = 3x + 1 is reflected about the y-axis,|
|then an equation of the reflection is y =|
|(A) 3x - 1|
|(B) log3 (x - 1)|
|(C) log3 (x + 1)|
|(D) 3-x + 1|
|(E) 1 - 3x|
|The answer is D.||The reflection of y = (x) in the y-axis is y = (-x)|
You should be familiar with the definitions and graphs of these trigonometric functions:
sine, cosine, tangent, cotangent, secant, and cosecant
|Exercise:||If (x) = sin(tan-1 x), what is the range of ?|
|(C) (0, 1]|
|(D) (-1, 1)|
|(E) [-1, 1]|
|The answer is D.||The range of sin x is (E), but the points at which sin x = 1 (/2 + k),|
|tan-1 x is undefined. Therefore, the endpoints are not included.|
Note: The range is expressed using interval notation:
Zeros of a Function
These occur where the function (x) crosses the x-axis. These points are also called the
roots of a function.
|Exercise:||The zeros of (x) = x3 - 2x2 + x is|
|(A) 0, -1|
|(B) 0, 1|
|(E) -1, 1|
|The answer is B.||(x) = x(x2 - 2x + 1) = x(x -1)2|
Properties of Graphs
You should review the following topics:
d) Relationships between the graph of
|y = (x) and||y = k(x)|
|y = (kx)|
|y - k = (x - h)|
|y = |(x)||
|y = (|x|)|