LESSON 8 FUNCTIONS
You should learn to:
1.
2.
3.
4.
Decide whether a relation between two variables is a function.
Find domains and ranges of functions.
Use function notation and evaluate functions.
Use functions to model and solve real-life problems.
Terms to know: relation, function, domain, range, independent variable, dependent variable, function notation,
piecewise-defined (or piecewise) function, implied domain versus specified domain
A relation is a set of ordered pairs.
A function f from a set A to a set B is a relation that assigns to each element (member) in the set A
exactly one element in the set B.
Set A is the domain (or set of inputs) and Set B is the range (or set of outputs) for the function f.
Example 1: Decide whether or not each relation is a function. For each function, list the domain and range.
b.
a.
 0, 2 , 1,5 ,  2,3 ,  2, 4
Function: No
The input of 2 has two different outputs; 3 and 4
c.
 0,0 , 1, 2 ,  2, 4 , 3,6
Function: Yes
Function: Yes
Domain: 1, 2,3, 4
Domain: 0,1, 2,3
Range: 1, 4,5
Range:
0, 2, 4, 6
Example 2: Decide whether each relation represents y (output) as a function of x (input).
y
a.
b.
Input,
2
2
2
1
2
3


x
Output,
x
y








No, the input (x value) of 2
has three different outputs (y values)
Yes, each input (x-value)
has only one output (y -value)

Functions are usually represented by equations. For example, y  2 x  1 represents the variable y as a function of
the variable x. x is the independent variable and y is the dependent variable. (The value of y depends upon
the chosen value for x). The domain is the set of all x values, and the range is the set of all y values.
Example 3: Which of the equations represents y as a function of x?
a. x 2  y  1
b.
y2  x  1
x x
x2  y  1
 x2
y2  x  1
 x2
y2  1 x
y  x 1
one equation
Yes, y is a function of x.
2
y2  1 x
y   1 x
y  1  x or y   1  x
two equations
No, y is not a function of x.
c. y  x  1
d.
x  y 1
y  x 1
x
x  y 1
x
x
y   x 1
x
y  1 x
one equation
y  1  x or y  (1  x )
Yes, y is a function of x.
two equations
No, y is not a function of x.
Domain: x  1
Function notation uses expressions such as f  x  , g  x  , and h  x  to represent y in the equation of a function.
For example, y  2 x  1 could be written as f  x   2 x  1 in function notation.
Note: f  x  does not mean “ f times x .”
Example 4: Let f ( x)   x 2  2 x  1 , g ( x)  3  2 x , h( x)  2 x 2 , and k ( x)  1 .
Evaluate (find) the following:
a.
f (3)
b.
g (3)
c. k (3)
d.
f (3)
f ( x)   x 2  2 x  1
g ( x)  3  2 x
k ( x)  1
f ( x)   x 2  2 x  1
f (3)  32  2 3  1
f (3)  9  6  1
g (3)  3  2 3
k (3)  1
g (3)  3  6
f (3)  (3) 2  2(3)  1
f (3)  9  6  1
f (3)  2
g (3)  3
f (3)  14
g (3)  DNE
e. h(3)
g. f ( )
f. k (3)
h. h( x  1)
h( x )  2 x 2
k ( x)  1
f ( x)   x 2  2 x  1
h( x )  2 x 2
h( 3)  2( 3) 2
h(3)  2 9
k (3)  1
f ( )   2  2  1
h( x  1)  2( x  1) 2
h( x  1)  2( x  1)( x  1)
h( x  1)  2( x 2  2 x  1)
h( 3)  18
h( x  1)  2 x 2  4 x  2
i. k ( x 2  1)
k ( x)  1
k ( x 2  1)  1
A piecewise-defined (piecewise) function is a function that is defined by two or more equations over a
specified domain.
2  x , x <  1
 2
Example 5: If f ( x)   x ,  1  x  1
x ,
x>1

a.
f (2)
b.
f ( 1)
c.
f (0)
f (2)  2  (2)
2  x , x <  1

f ( x)   x 2 ,  1  x  1
x ,
x>1

f (1)  (1) 2
2  x , x <  1

f ( x)   x 2 ,  1  x  1
x ,
x>1

f (0)  02
f (2)  4
f (1)  1
f (0)  0
2  x , x <  1

f ( x)   x 2 ,  1  x  1
x ,
x>1

d.
find:
f (1)
e.
f (2)
2  x , x <  1

f ( x)   x 2 ,  1  x  1
x ,
x>1

2  x , x <  1

f ( x)   x 2 ,  1  x  1
x ,
x>1

f (1)  12
f (2)  2
f (1)  1
f.
Graph the piecewise function f .
Example 6: What is the domain for each of the functions below?
a. f ( x)   x 2  2 x  1
c. f ( x) 
b. g ( x)  3  2 x
3  2x  0
All Real Numbers:
1
x  5x  6
2
x2  5x  6  0
3  3
2 x  3
2 x 3

2 2
3
x
2
( x  6)( x  1)  0
x  6, x  1
Domain Restrictions: 1. Fractions (no zero denominators)
2. Even Roots (no even roots of negative numbers)
Sometimes, it is easier to find the domain and range of a function by looking at its graph.
Example 7: Find the domain and range from graphs:
a.
y
f ( x)   x 2  4
b. g ( x) 









x





y
1
x

















x


Domain : x  2 or x  2
Domain : x  0
Range : y  0
Range : y  0
ASSIGNMENT 8
Pages 24-29: (1-6, 10-13, 16, 22, 26, 32, 38, 46, 47, 56, 57, 63, 68, 78, 94, 99);
Page 12: (28, 55); Page A34: (62); Page A61(103, 158) Page 14 ( 81, 82)
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