Functions and their Graphs
Definition of a Function
A function is a correspondence between two sets where
each element in the first set, called the domain, corresponds
to exactly one element in the second set, called the range.
All functions are relations but not all relations are functions.
Given the function f (x) = 2x3 − 3x2 + 6,
find f (−1).
Solution:
Consider the independent variable x to be a
placeholder.
To find f (−1), substitute x = −1 into the function.
f (−1) = 2(−1)3 − 3(−1)2 + 6
= −2 − 3 + 6
=1
f(-1) = 1

For the given function f (x) = x2 − 3x, evaluate f (x
+ 1).

Replace the argument x with a placeholder.

f(□) = (□)2 – 3(□)

Substitute x + 1 for the argument.

f(x + 1) = (x + 1)2 – 3(x + 1)

Eliminate the parentheses.

f(x + 1) = x2 + 2x + 1 -3x – 3

Combine like terms.

f(x + 1) = x2 – x - 2


The domain is the set of all inputs (x-values) and
the range is the set of all corresponding outputs
(y-values).
An explicit domain is stated, whereas an implicit
domain is found by
excluding x-values that:
◦ make the function undefined
(such as denominator = 0)
◦ result in a nonreal output
(even roots of negative real numbers)
The Square Function
The Square Root Function
The Cube Function
The Cube Root Function
The Absolute Value
Function
The Reciprocal Function
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When f(-x) = f(x) we say that f(x) is an even
function.
Consider f(x) = x2 – 3
f(-x) = (-x)2 – 3 = x2 – 3 = f(x)
When f(-x) = - f(x) we say that f(x) is an odd
function.
Consider g(x) = x5 + x3
g(-x) = (-x)5 + (-x) 3 = -x5 –x3 = -g(x)
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Decreasing
(-∞, -2) ∪(0, 2) ∪(6,∞)
Increasing
(-2, -1) ∪ (2,6)
Constant
(-1,0)
The slope of the secant line is used to represent the
average rate of change of the function.
Average
rate of
change

For the function f (x) = x2 − x, find

Solution:
y
8
x

f ( x)   x 2
2 x

x0
0 x 2
x2
6
4
2
0
x
2
4
4
2
0
x
2
4
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Vertical shifts: (c>0)
f(x) + c shift c units up
f(x) – c shift c units down
Horizontal shifts: (c>0)
f(x + c) shift c units left
f(x – c) shift c units right




The graph of –f(x) is obtained by
reflecting the graph of f(x) about
the x-axis
f(x) = x2
g(x) = -x2
The graph of f(-x) is obtained by
reflecting the graph of f(x) about
the y-axis
The graph of cf(x) is found by:
Vertically stretching the graph
of f(x) if c > 1
Vertically compressing the
graph of f(x) if 0<c<1
The graph of f(cx) is found by:
Horizontally stretching the
graph of f(x) if 0<c<1
f ( x)
f (2x)
Horizontally compressing the
graph of f(x) if c>1
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Sum Function
(f + g)(x) = f (x) + g(x)
{domain of f } ∩ {domain of g}
Difference Function (f − g)(x) = f (x) − g(x)
{domain of f } ∩ {domain of g}
Product Function
(f · g)(x) = f (x) · g(x)
{domain of f } ∩ {domain of g}
Quotient Function
{domain of f } ∩ {domain of g} ∩ {g(x) ≠ 0}
y
8
6
x
 2
f ( x)   x
2 x

x0
4
0 x 2
2
x2
0
x
2
4
4
2

x

f  1( x)   x
x

2
0
2
x
x0
0 x4
x4
4