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

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4-2

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

  2. y = y + 5 x + 7 8 3 y = ± x Warm Up Solve for y. 1.x = 3y –7 y = 8x – 5 2.x = 3.x = 4 – y y = 4 – x 4.x = y2

  3. Objectives Graph and recognize inverses of relations and functions. Find inverses of functions.

  4. Vocabulary inverse relation inverse function

  5. The multiplicative inverse of 5 is The multiplicative inverse matrix of You have seen the word inverse used in various ways. The additive inverse of 3 is –3.

  6. Remember! 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. 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.

  7. Example 1: Graphing Inverse Relations Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Graph each ordered pair and connect them. ● Switch the x- and y-values in each ordered pair. ● ● ●

  8. 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. Check It Out! Example 1 Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair. • • • • •

  10. Check It Out! Example 1 Continued 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}

  11. 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. 1 is subtracted from the variable, x. 2 1 1 1 1 f(x) = x – Add to x to write the inverse. 2 2 2 2 f–1(x) = x + Example 2: Writing Inverses of by Using Inverse Functions Use inverse operations to write the inverse of f(x) = x – if possible.

  13. 1 1 1 = 1 2 2 2 1 1 1 1 f(x) = x – f(1) = 1 – 2 2 2 2 2 f–1(x) = x + Substitute for x. f–1( ) = + The inverse function does undo the original function.  Example 2 Continued Check Use the input x = 1 in f(x). Substitute 1 for x. Substitute the result into f–1(x) = 1

  14. x 3 x 3 Check It Out! Example 2a Use inverse operations to write the inverse of f(x) =. f(x) = The variable x, is divided by 3. f–1(x) = 3x Multiply by 3 to write the inverse.

  15. 1 x 3 3 1 1 1 = 1 3 3 3 3 f–1( ) = 3( ) Substitute for x. The inverse function does undo the original function.  Check It Out! Example 2a Continued CheckUse the input x = 1 in f(x). f(x) = f(1) = Substitute 1 for x. Substitute the result into f–1(x) f–1(x) = 3x = 1

  16. is added to the variable, x. f(x) = x + 2 2 2 2 2 3 3 3 3 3 Subtract from x to write the inverse. f–1(x) = x – Check It Out! Example 2b Use inverse operations to write the inverse of f(x) = x + .

  17. f(1) = 1 + 5 5 5 = 5 3 3 3 2 2 2 2 f(x) = x + 3 3 3 3 3 Substitute for x. f–1(x) = x – f–1( ) = – The inverse function does undo the original function.  Check It Out! Example 2b Continued CheckUse the input x = 1 in f(x). Substitute 1 for x. Substitute the result into f–1(x) = 1

  18. Helpful Hint The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses Undo operations in the opposite order of the order of operations.

  19. 1 1 3 3 f–1(x) = x + 7 f–1(6) = (6) + 7= 2 + 7= 9  Example 3: Writing Inverses of Multi-Step 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. f(x) = 3(x – 7) First, undo the multiplication by dividing by 3. Then, undo the subtraction by adding 7. Check Use a sample input. f(9) = 3(9 – 7) = 3(2) = 6

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

  21. You can also find the inverse function by writing the original function with x and y switched and then solving for y.

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

  23. f f –1 Example 4 Continued f–1(x) = –2(x + 5) Set y = f(x). f–1(x) = –2x – 10 Simplify. Then graph f–1.

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

  25. f–1(x) = 3 2 f –1 f x – 3 Check It Out! Example 4 Set y = f(x). Then graph f–1.

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

  27. 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}

  28. f f–1 1 4 3 3 f–1(x) = x + Lesson Quiz: Part II 2. Graph f(x) = 3x – 4. Then write and graph the inverse.

  29. 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. F = C + 32; 77° F 5 9 9 5 Lesson Quiz: Part III

  30. HW: Text Pg 245 Tuesday (1/13) #4-16 Wednesday (1/14) # 20 – 40 even, 41 – 46 all

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