1. Inverse functions and Relations
6.2
Inverse functions and Relations

2. Recall that a relation is a set of ordered pairs.
The inverse relation is the set of ordered pairs obtained by exchanging the coordinates of each ordered pair.
The domain of a relation becomes the range of its inverse
The range of a relation becomes the domain of its inverse.

3. Example 1 The ordered pairs of the relation
{(1, 3), (6, 3), (6, 0), (1, 0)} are the coordinates
of the vertices of a rectangle. Find the inverse
of this relation. Describe the graph of the inverse.

4. Example 2 The ordered pairs of the relation
{(–3, 4), (–1, 5), (2, 3), (1, 1), (–2, 1)} are the coordinates of the vertices of a pentagon. What is the inverse of this relation?

5. Inverse Properties
As with relations, the ordered pairs of inverse functions are also related. We write the inverse of the function f(x) as f-1(x)

6. Example 3a: Given the following table, find f –1(9). x f(x) 4 7 3 9 62 10

7. Example 3b: Given the following table, find f –1(6). x f(x) 4 7 3 9 62 10

8. Graphs of inverses
When the inverse of a function is also a function, the original function is one–to–one. Since we use a vertical line test to determine if a relation is a function, we can use a horizontal line test to determine whether the inverse is also a function.

9. Example 4
Use the horizontal line test to determine whether the inverse of each function is also a function.
a) f(x) = 2x2
b) h(x) = -4x + 7

10. To find an inverse
To find an inverse, exchange the domain and the range.
(ie switch the x and y and solve for y)

11. Example 5
Find the inverse of 𝒇 𝒙 =− 𝟏 𝟐 𝒙+𝟏.

12. Example 6
Find the inverse of 𝒇 𝒙 = 𝒙 𝟐 +𝟒. Then graph the function and its inverse.

13. Example 7
Find the inverse of 𝒇 𝒙 = 𝟐𝒙 −𝟏.

14. Example 8

15. Example 9

16. The graph of f(x) is shown below. Find f-1(3).
Example 10
The graph of f(x) is shown below. Find f-1(3).

17. Example 11
Alex began a new exercise routine. To gain the maximum benefit from his exercise, Alex calculated his maximum target heart rate using the function T = 0.85(220 – A), where A represents his age and T represents his maximum target heart rate.
A) Find the inverse of this function.
B) Use the inverse to find his age if the maximum heart rate
is 152.