2. Example 1A: Using the Horizontal-Line Test
Use the horizontal-line test to determine whether the inverse of the blue relation is a function.
The inverse is a function because no horizontal line passes through two points on the graph.

3. Example 1B: Using the Horizontal-Line Test
Use the horizontal-line test to determine whether the inverse of the red relation is a function.
The inverse is a not a function because a horizontal line passes through more than one point on the graph.

4. Check It Out! Example 1
Use the horizontal-line test to determine whether the inverse of each relation is a function.
The inverse is a function because no horizontal line passes through two points on the graph.

5. Recall from Lesson 7-2 that to write the rule for the inverse of a function, you can exchange x and y and solve the equation for y. Because the value of x and y are switched, the domain of the function will be the range of its inverse and vice versa.

6. Example 2: Writing Rules for inverses
Find the inverse of Determine whether it is a function, and state its domain and range.
Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

7. Example 2 Continued
Step 1 Find the inverse.
Rewrite the function using y instead of f(x).
Switch x and y in the equation.
Cube both sides.
Simplify.
Isolate y.

8. Example 2 Continued
Because the inverse is a function,
The domain of the inverse is the range of f(x):{x|x R}.
The range is the domain of f(x):{y|y R}.
Check Graph both relations to see that they are symmetric about y = x.

9. Check It Out! Example 2
Find the inverse of f(x) = x3 – 2. Determine whether it is a function, and state its domain and range.
Step 1 The horizontal-line test shows that the inverse is a function. Note that the domain and range of f are all real numbers.

10. Check It Out! Example 2 Continued
Step 1 Find the inverse.
y = x3 – 2
Rewrite the function using y instead of f(x).
x = y3 – 2
Switch x and y in the equation.
x + 2 = y3
Add 2 to both sides of the equation.
Take the cube root of both sides. 3
x + 2 = y 3
x + 2 = y
Simplify.

11. Check It Out! Example 2 Continued
Because the inverse is a function,
The domain of the inverse is the range of f(x): R.
The range is the domain of f(x): R.
Check Graph both relations to see that they are symmetric about y = x.

12. You can use composition of functions
to verify that 2 functions are inverses.
The inverse functions “undo” each other,
When you compose two inverses…
the result is the input value of x.

13. Then f(x) and g(x) are inverse functions
If f(g(x)) = g(f(x)) = x
Then f(x) and g(x) are inverse functions
Example 1:
Because f(g(x)) = g(f(x)) = x, they are inverses.

14. Find the composition f(g(x)).
Determine by composition whether each pair of functions are inverses.
Example 2: 1 3
f(x) = 3x – 1 and g(x) = x + 1
Find the composition f(g(x)).
Substitute x + 1 for x in f. 1 3
Use the Distributive Property.
= (x + 3) – 1
f(g(x)) = x + 2
Simplify.
The functions are NOT inverses.

15. Because f(g(x)) = g(f(x)) = x, they are inverses.
Example 3
Determine by composition whether each pair of functions are inverses. 3 2
f(x) = x + 6 and g(x) = x – 9
Find the composition f(g(x)) and g(f(x)).
= x –
= x + 9 – 9
Because f(g(x)) = g(f(x)) = x, they are inverses.

16. Find the compositions f(g(x)) and g(f(x)).
Example 4
f(x) = x2 + 5 and for x ≥ 0
Find the compositions f(g(x)) and g(f(x)).
Substitute for x in f.
Simplify.
= x 10 x -
Because f(g(x)) ≠ x, f (x) and g(x) are NOT inverses.

17. Are the following functions Inverses?
Example 5:
Are the following functions Inverses?
For x ≠ 1 or 0, f(x) = and g(x) = 1 x
x – 1
= (x – 1) + 1
Because f(g(x)) = g(f (x)) = x f(x) and g(x) are inverses.

18. Lesson Quiz: Part I
1. Use the horizontal-line test to determine whether the inverse of each relation is a function.
A: yes; B: no

19. Lesson Quiz: Part II
2. Find the inverse f(x) = x2 – 4. Determine whether it is a function, and state its domain and range.
not a function
D: {x|x ≥ 4}; R: {all Real Numbers}

20. Lesson Quiz: Part III
3. Determine by composition whether f(x) = 3(x – 1)2 and g(x) = are inverses for x ≥ 0.
yes