1. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions 4-2 Inverses of Relations and Functions Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz Holt McDougal Algebra 2

2. 4-2 Inverses of Relations and Functions Warm Up Solve for y. 1. x = 3y –7 4. x = y 2 3. x = 4 – y 2. x = y + 5 8 x + 7 3 y = y = 8x – 5 y = 4 – x y = ± x

3. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Graph and recognize inverses of relations and functions. Find inverses of functions. Objectives

4. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions inverse relation inverse function Vocabulary

5. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions You have seen the word inverse used in various ways. The additive inverse of 3 is –3. The multiplicative inverse of 5 is The multiplicative inverse matrix of

6. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y- values in each ordered pair of the relation. A relation is a set of ordered pairs. A function is a relation in which each x-value has, at most, one y-value paired with it. Remember!

7. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Example 1: Graphing Inverse Relations x 0158 y 2569 Graph each ordered pair and connect them. x 2569 y 0158 ● ● ● ● Switch the x- and y-values in each ordered pair.

8. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Example 1 Continued Reflect each point across y = x, and connect them. Make sure the points match those in the table. Domain:{x|0 ≤ x ≤ 8} Range :{y|2 ≤ x ≤ 9} Domain:{x|2 ≤ x ≤ 9} Range :{y|0 ≤ x ≤ 8}

9. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 13456 y 01235 Check It Out! Example 1 Graph each ordered pair and connect them. x 01235 y 13456 Switch the x- and y-values in each ordered pair.

10. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Reflect each point across y = x, and connect them. Make sure the points match those in the table. Domain:{1 ≤ x ≤ 6} Range :{0 ≤ y ≤ 5} Domain:{0 ≤ y ≤5} Range :{1 ≤ x ≤ 6} Check It Out! Example 1 Continued

11. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions When the relation is also a function, you can write the inverse of the function f(x) as f –1 (x). This notation does not indicate a reciprocal. Functions that undo each other are inverse functions. To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f –1 (x).

12. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Use inverse operations to write the inverse of f(x) = x – if possible. Example 2: Writing Inverses of by Using Inverse Functions 1 2 is subtracted from the variable, x. 1 2 Add to x to write the inverse. 1 2 f(x) = x – 1 2 1 2 f –1 (x) = x +

13. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Example 2 Continued Substitute 1 for x. Check Use the input x = 1 in f(x). f(x) = x – 1 2 f(1) = 1 – 1 2 1 2 Substitute for x. Substitute the result into f –1 (x) 1 2 f –1 (x) = x + = 1 = 1 2 1 2 f –1 ( ) = + 1 2 1 2 The inverse function does undo the original function.

14. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Use inverse operations to write the inverse of f(x) =. x 3 Check It Out! Example 2a f –1 (x) = 3x f(x) = The variable x, is divided by 3. Multiply by 3 to write the inverse. x 3

15. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Substitute 1 for x. Check Use the input x = 1 in f(x). 1 3 Substitute for x. Substitute the result into f –1 (x) = 1 = 1 3 f –1 ( ) = 3( ) 1 3 1 3 The inverse function does undo the original function. Check It Out! Example 2a Continued f(x) = x 3 f(1) = 1 3 f –1 (x) = 3x

16. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Use inverse operations to write the inverse of f(x) = x +. Check It Out! Example 2b is added to the variable, x. 2 3 Subtract from x to write the inverse. 2 3 f(x) = x + 2 3 2 3 f –1 (x) = x – 2 3

17. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Check It Out! Example 2b Continued Substitute 1 for x. Check Use the input x = 1 in f(x). f(1) = 1 + 2 3 5 3 Substitute for x. Substitute the result into f –1 (x) 2 3 f –1 (x) = x – = 1 = 5 3 2 3 f –1 ( ) = – 5 3 5 3 The inverse function does undo the original function. f(x) = x + 2 3

18. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Undo operations in the opposite order of the order of operations. The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses Helpful Hint

19. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Use inverse operations to write the inverse of f(x) = 3(x – 7). The variable x is subtracted by 7, then is multiplied by 3. First, undo the multiplication by dividing by 3. Then, undo the subtraction by adding 7. Example 3: Writing Inverses of Multi-Step Functions f(x) = 3(x – 7) 1 3 f –1 (x) = x + 7 Check Use a sample input. f(9) = 3(9 – 7) = 3(2) = 6 1 3 f –1 (6) = (6) + 7= 2 + 7= 9

20. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Use inverse operations to write the inverse of f(x) = 5x – 7. The variable x is multiplied by 5, then 7 is subtracted. First, undo the subtraction by adding by 7. Then, undo the multiplication by dividing by 5. f(x) = 5x – 7. Check Use a sample input. Check It Out! Example 3 f –1 (x) = x + 7 5 f –1 (3) = = = 2 f(2) = 5(2) – 7 = 3 10 5 3 + 7 5

21. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions You can also find the inverse function by writing the original function with x and y switched and then solving for y.

22. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Example 4: Writing and Graphing Inverse Functions Switch x and y. Solve for y. Set y = f(x) and graph f. Graph f(x) = – x – 5. Then write the inverse and graph. 1 2 1 2 y = – x – 5 1 2 x = – y – 5x + 5 = – y 1 2 –2x – 10 = y Write in y = format. y = –2(x + 5)

23. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Example 4 Continued Set y = f(x). f –1 (x) = –2(x + 5) Simplify. Then graph f –1. f –1 (x) = –2x – 10 f f –1

24. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Switch x and y. Solve for y. Set y = f(x) and graph f. Graph f(x) = x + 2. Then write the inverse and graph. 2 3 x – 2 = y 2 3 3x – 6 = 2y Write in y = format. Check It Out! Example 4 2 3 x = y + 2 2 3 y = x + 2 x – 3 = y 3 2

25. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Set y = f(x). Then graph f –1. f f –1 Check It Out! Example 4 f –1 (x) = x – 3 3 2

26. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Anytime you need to undo an operation or work backward from a result to the original input, you can apply inverse functions. In a real-world situation, don’t switch the variables, because they are named for specific quantities. Remember!

27. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Lesson Quiz: Part I 1. A relation consists of the following points and the segments drawn between them. Find the domain and range of the inverse relation: D:{x|1  x  8}R:{y|0  y  9} x03469 y12578

28. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Lesson Quiz: Part II 2. Graph f(x) = 3x – 4. Then write and graph the inverse. f –1 (x) = x + 1 3 4 3 f f –1

29. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions Lesson Quiz: Part III 3. A thermometer gives a reading of 25° C. Use the formula C = (F – 32). Write the inverse function and use it to find the equivalent temperature in °F. 5 9 F = C + 32; 77° F 9 5

30. Holt McDougal Algebra 2 4-2 Inverses of Relations and Functions HW: Text Pg 245 Tuesday (1/13) #4-16 Wednesday (1/14) # 20 – 40 even, 41 – 46 all