Finding Inverse Functions Guided Notes. Example 1: Find the inverse of a(b) = 3b 7

Transcription

1 Inverse Functions Finding Inverse Functions Guided Notes We ve explored the idea that the inverse of a function undoes the function. One way of finding the inverse of a function is to. You can think of this as switching the inputs and the outputs. Example 1: Find the inverse of a(b) = 3b 7 Example : Find the inverse of a(b) = 3b 3 b + 5 Example 3: Find the inverse of a(b) = x + 3x + But the function in example 3 is not so even though we can find an inverse, that inverse is not a function. We can solve this problem by of both the function and its inverse. Math Week Packet Page 1

2 Inverse Functions Similarly, we can also easily find the inverse of the graph of a function by switching the inputs and the outputs. In graph form this looks like. Example : Sketch the inverse of the graph shown below. 5 5 Example 5: Sketch the inverse of the graph shown below. 5 5 Is the inverse a function? Why or why not? Math Week Packet Page

3 Inverse Functions Example 6: Sketch the inverse of the graph shown below. 5 5 Example 7: Sketch the inverse of the graph shown below. 5 5 Generalize: To sketch the graph of the inverse of a function, reflect the graph of the function across the line. This will switch the inputs and the outputs. Groupwork: In your groups, read and complete the work in the textbook beginning with # on page 301 and ending with # on page 30. Homework: Pages # 3, (a- f) (use your calculator), 5, and 7. The following problems are reach problems: 3c, 3e, e, and f. All the rest are standard problems. Math Week Packet Page 3

4 Do Now 11.5 Do Now 11.5 Evaluating Functions Consider the function f(x) = 3x 3 x + 5x. 1. Find f(1). Find f() 3. Find f(10). Find f(f(3) 5. Find f - 1 (1) 6. Find f - 1 (f - 1 (3108) 7. Find f(f(15) 8. Find f(0) Math Week Packet Page

5 Do Now 1 Do Now 1 Inverses of Functions 1. Find inverse functions for these functions. a. k(x) = (x + 6) 3 b. v(x) = 15 x. Sketch the graph of the inverse of each function on the same axes. Determine whether the inverse is a function or merely a relation. a. b Is the inverse a function? Why or why not? Is the inverse a function? Why or why not? Math Week Packet Page 5

6 Students will be able to (SWBAT): Write an explicit and a recursive function rule for a linear table of values. Describe in words or write a recursive function rule for a non-linear table of values. Determine whether a relation is a function based on a description, a table, or a graph. Find domain and range of a function from a graph. Based on an equation, determine domain restrictions for rational and radical functions. Find the inverse of a linear function. Given the graph of a linear function, graph its inverse. Evaluate compositions of functions, and find a rule for a composition. 1. Consider the functions f(x)=3x+7 g(x)= x +1 Calculate each value a. f() b. g(3) c. f(g(5)) d. g(f()) e. f () g(3) f. f() + g(3) Math Week Packet Page 6

7 . Given the functions R(x) = x Q(x)= x + x +1 a. Find a formula for R(Q(x)) b. Find a formula for Q(R(x)) 3. State the domain of each function. You must write inequalities or use interval notation a. b(x) = x 6 b. c(x) = x 6 Domain: Domain: c. d(x) = 8 + x d. e(x) = 1+ x 1+ x Domain: Domain: Math Week Packet Page 7

8 . State the domain and range of each function. The functions do NOT extend beyond the graphs shown.. a. b. Domain: Domain: Range: Range: Math Week Packet Page 8

9 5. a. Complete the difference column for this table. b. Find a recursive function definition and an explicit (closed form) function definition that agree with the table below. Input, n Output, p(n) 0-8 Δ Recursive definition: p(n) = " # $ Explicit definition: p(n) = Math Week Packet Page 9

10 6. For each function: a. Sketch a graph of the function below by making a table of values or using your graphing calculator. b. On the same graph, graph the inverse function or relation. Label both graphs to distinguish them. c. Is the inverse a function or merely a relation? d. Find the inverse function or relation. f(x)= x + 3 g(x)= 8x h(x)= 0.5x j(x)= 3x i. f(x)= x + 3 ii. g(x)= 8x Graph f(x) and its inverse. Graph g(x) and its inverse. Is the inverse a function? Write the inverse using the correct name for it, in this case f 1 (x). Is the inverse a function? Write the inverse using the correct name for it. Math Week Packet Page 10

11 iii. h(x)= 0.5x iv. j(x)= 3x Graph h(x) and its inverse. Graph j(x) and its inverse. Is the inverse a function? Write the inverse using the correct name for it. Is the inverse a function? Write the inverse using the correct name for it. Math Week Packet Page 11

12 7. a. Which of the following recursive functions best fits the table below? # if n = 0 i. a(n) = $ % a(n 1) 3 if n > 0 Input, n $ if n = 0 ii. a(n) = % & a(n 1) 5 if n > 0 # if n = 0 iii. a(n) = $ % a(n 1) + n 5 if n > 0 # if n = 0 iv. a(n) = $ % a(n 1) n if n > 0 Output, a(n) b. Fill in the value for a(5) in the table using the definition you chose Write an equation for a linear function with a slope of and passing through the point (5, - ). Math Week Packet Page 1

13 Week Assignment Guide Monday Tuesday Thursday B- Block Do Now 11: Functions Pop Quiz In Class: Inverse Functions, Graphically and Algebraically. Homework: Pages # 3, (a- f) (use your calculator), 5, and 7. The following problems are reach problems: 3c, 3e, e, and f. All the rest are standard problems. Do Now: DN 11.5 Evaluating Functions In Class: Finding Zeros with Calculators, Evaluating Functions with Calculators, Begin Review if Time Homework: None DN 1 Inverses of Functions In Class: Test Review Homework: Finish Test Review C- Block Do Now 11: Functions Pop Quiz In Class: Inverse Functions, Graphically and Algebraically. Homework: Pages # 3, (a- f) (use your calculator), 5, and 7. The following problems are reach problems: 3c, 3e, e, and f. All the rest are standard problems. DN 1 Inverses of Functions In Class: Test Review Homework: Finish Test Review Friday In Class: Unit 1 Test Homework: None In Class: Unit 1 Test Homework: None