4.2 Functions and Their Inverses
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities, graph equations and/or inequalities on coordinate axes with labels and
scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations or
inequalities, and interpret solutions as viable or nonviable options in a modeling
situation.
For the Board: You will be able to determine whether the inverse of a function is a function, and write
rules for the inverses of functions.
Anticipatory Set:
The inverse of a function f(x) “undoes” f(x).
f(x)
Inverse of f(x)
3
5 5
3
(3, 5)
(5, 3)
The ordered pairs are an interchange of x and y.
Open the book to page 242 and read example 1.
Example: (1) Graph the relation and connect the points.
(2) Determine the domain and range.
(3) Determine the ordered pairs of the inverse.
(4) Graph the inverse.
(5) Determine the domain and range of the inverse.
x
y
0
2
1
5
Domain: 0 < x < 8
x
y
2
0
Domain: 2 < x < 9
5
6
Range:
5
1
6
5
Range:
8
9
2<y<9
9
8
0<y<8
The graph of the inverse is a reflection of the graph of the function across the line y = x.
The inverse may or may not be a function.
Graphing Activity:
Practice: (1) Graph the relation and connect the points.
(2) Determine the domain and range.
(3) Determine the ordered pairs of the inverse.
(4) Graph the inverse.
(5) Determine the domain and range of the inverse.
x
y
1
0
3
1
Domain: 1 < x < 6
x
y
0
1
1
3
Domain: 0 < x < 5
4
2
5
3
Range:
2
4
Range:
6
5
0<y<5
3
5
5
6
1<y<6
Vertical Line Test
If any vertical line passes through more than one point on the graph of a relation, the relation is
not a function.
Horizontal Line Test
If any horizontal line passes through more than one point on the graph of a relation, the inverse
relation is not a function.
Examples:
The relation passes the vertical line test, therefore
it is a function.
The relation passes the horizontal line test,
therefore its inverse is a function.
The relation passes the vertical line test, therefore
it is a function.
The relation fails the horizontal line test, therefore
It’s inverse is not a function.
Open the book to page 450 and read example 1.
Example: Use the vertical line test to determine whether the graph is a function.
Use the horizontal-line test to determine whether the inverse of the graph is a function.
a.
b.
The relation fails the vertical line test
and therefore is not a function, but it
passes the horizontal line test, therefore
it’s inverse is a function.
The relation passes the vertical line test
and therefore is a function, but it fails
the horizontal line test, therefore it’s
inverse is not a function.
White Board Activity:
Practice: Use the vertical line test to determine whether the graph is a function.
Use the horizontal-line test to determine whether the inverse of the graph is a function.
The relation passes the vertical line test
and therefore is a function.
The relation passes the horizontal line test
and therefore it’s inverse is also a function
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 245 – 246 prob. 2, 3, 18, 19.
Text: pgs. 453 – 454 prob. 1 – 3, 9 – 11.
For a Grade:
Text: pg. 245 prob. 18; pg. 453 prob. 10