Determine whether the inverse of a function is a function.

1. Objectives Determine whether the inverse of a function is a function. Write rules for the inverses of functions.

2. Vocabulary one-to-one function When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value.

3. Recall that the vertical-line testcan help you determine whether a relation is a function. Vertical Line Test:a relation is a function if a vertical line drawn through its graph, passes through only one point. Similarly, the horizontal-line test can help you determine whether the inverse of a function is a function.

4. 1-6 Relations and Functions Vertical Line Test Would this graph be a function? YES

5. 1-6 Relations and Functions Vertical Line Test Would this graph be a function? NO

6. Use the horizontal line test to decide if the inverse is a function.

7. 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.

8. 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 afunction because a horizontal line passes through more than one point on the graph.

9. 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.

10. 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.

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

12. 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.

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

14. 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 theinverse is a function. Note that the domain and range of f are all real numbers.

15. 3 3 x + 2 = y 3 3 x + 2 = y 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. Simplify.

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

17. You have seen that the inverses of functions are not necessarily functions. When both a relation and its inverses are functions, the relation is called a one-to-one function. In a one-to-one function, each y-value is paired with exactly one x-value. You can use composition of functions to verify that two functions are inverses. Because inverse functions “undo” each other, when you compose two inverses the result is the input value x.

19. f(g(x)) = 3( x + 1) – 1 Substitute x + 1 for x in f. 1 1 1 3 3 3 Example 3: Determining Whether Functions Are Inverses Determine by composition whether each pair of functions are inverses. f(x) = 3x – 1 and g(x) = x + 1 Find the composition f(g(x)). Use the Distributive Property. = (x + 3) – 1 = x + 2 Simplify.

20. Example 3 Continued Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)). CheckThe graphs are not symmetric about the line y = x.

21. For x ≠ 1 or 0, f(x) = and g(x) = + 1. 1 1 x x –1 Example 3B: Determining Whether Functions Are Inverses Find the compositions f(g(x)) and g(f (x)). = (x –1) + 1 = x = x Because f(g(x)) = g(f (x)) = x for all x but 0 and 1, f and g are inverses.

22. Example 3B Continued CheckThe graphs are symmetric about the line y = x for all x but 0 and 1.

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

24. Check It Out! Example 3a Continued Check The graphs are symmetric about the line y = x for all x.

25. Substitute for x in f. f(g(x)) = + 5 = x + 25 +5 - 10 x = x – 10 x + 30 Check It Out! Example 3b f(x) = x2 + 5 and for x ≥ 0 Find the compositions f(g(x)) and g(f(x)). Simplify.

26. Check It Out! Example 3b Continued Because f(g(x)) ≠ x, f and g are not inverses. There is no need to check g(f(x)). Check The graphs are not symmetric about the line y = x.

27. 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

28. 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}

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