Properties of Functions
A function ƒ 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 ƒ is the set x.
The range of ƒ is the set y.
Combinations of Functions
If ƒ(x) = 3x + 1 and g(x) = x2 - 1
a) the sum ƒ(x) + g(x) = (3x + 1) + (x2 - 1) = x2 + 3x
b) the difference ƒ(x) - g(x) = (3x + 1) - (x2 - 1) = -x2 + 3x + 2
c) the product ƒ(x)g(x) = (3x + 1)(x2 - 1) = 3x3 + x2 - 3x - 1
d) the quotient ƒ(x)/g(x) = (3x + 1)/(x2 - 1)
e) the composite (ƒ ° g)(x) = ƒ(g(x))
= 3(x2 - 1) + 1 = 3x2
- 2
Inverse Functions
Functions ƒ and g are inverses of each other if
ƒ(g(x)) = x for each x in the domain of g
g(ƒ(x)) = x for each x in the domain of ƒ
The inverse of the function ƒ is denoted ƒ-1.
To find ƒ-1, switch x and y
in the original equation and solve the equation for y in
terms of x.
Exercise: | If ƒ(x) = 3x + 2, then ƒ-1(x) = |
(A) | |
(B) - 2 | |
(C) 3x - 2 | |
(D) x + 3 | |
(E) | |
The answer is E. | x = 3y + 2 |
3y = x - 2 | |
y = |
Even and Odd Functions
The function y = ƒ(x) is even if ƒ(-x) = ƒ(x).
Even functions are symmetric about the y-axis (e.g. y = x2)
The function y = ƒ(x) is odd if ƒ(-x) = -ƒ(x).
Odd functions are symmetric about the origin (e.g. y =
x3)
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 = ƒ(x) in the y-axis is y = ƒ(-x) |
Periodic Functions
You should be familiar with the definitions and graphs of these trigonometric functions:
sine, cosine, tangent, cotangent, secant, and cosecant
Exercise: | If ƒ(x) = sin(tan-1 x), what is the range of ƒ? |
(A) (-/2,/2) | |
(B) [-/2,/2] | |
(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 ƒ(x) crosses the x-axis. These points are also called the
roots of a function.
Exercise: | The zeros of ƒ(x) = x3 - 2x2 + x is |
(A) 0, -1 | |
(B) 0, 1 | |
(C) -1 | |
(D) 1 | |
(E) -1, 1 | |
The answer is B. | ƒ(x) = x(x2 - 2x + 1) = x(x -1)2 |
Properties of Graphs
You should review the following topics:
a) Intercepts
b) Symmetry
c) Asymptotes
d) Relationships between the graph of
y = ƒ(x) and | y = kƒ(x) |
y = ƒ(kx) | |
y - k = ƒ(x - h) | |
y = |ƒ(x)| | |
y = ƒ(|x|) |