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Exploration 5 will hand you a function. Walk around the room and pair up with someone. You choose a number and plug that n into their function. Then they will give you their result and you plug that result into your function. Do you notice anything? Now do it again but have them choose the first number. When you find someone whose function, when composed with yours, gives you an interesting result, pair up and complete the information below. Your function: Partner's function: X Original x 0 9 16 - 1 1 4 y Your function's output X Your function's output Your partner's function's output What relationship does your function have to your partner's function? Draw maps of what your function does to x using the flowchart below. For example, for f(x) = 3x + 2: X 3 +2 3x 3x + 2 EXAMPLE Your Function 0 00 Partner's Function What do you notice? 10cuppose Bobert and Cindy have partner functions. If Bobert's function has an input of 3, the output is 5. What happens if 5 is input into Cindy's function? Bobert's function has input 2 and output 9 Bobert's function has input -5 and output 99 Bobert's function has Bobert's function has input 5 and output a Cindy's function would... input x and output y Cindy's function would... Cindy's function would... Cindy's function would... Let your function be f(x) and your partner's function be g(x). Compute each of the following: f(g(1)) = f(g(5)) = g (f (8) ) = g(f (TT ) ) = f(g(a)) = g (f ( b ) ) = Inverse Function - A function f(x) has an inverse f (x) if and only if Exploration 6 A. Find the inverse of f (x) = x - 6 B. Find the inverse of g(x) = 3x C. Find the inverse of h(x) = 4x - 1. Is it h (x) = * or h (x) = * + 1?Exploration 7 Mr. Ricci's brother, Alex, used to work as a lube tech at a JiffyLube (where you can get your car's oil changed). He was paid $14 per hour, plus $2 per car he worked on. A. If he worked 40 hours a week, write an equation to represent his weekly pay f(x) based on the number of cars he worked on (x). B. Find the inverse function. What does each variable in the inverse function represent? C. How many cars did Alex need to work on to get $550 per week? D. How many cars did Alex work on if he made $566 ? Practice 8 For each pair of functions below, show they are inverses. A. f (x) = x +1;g(x) = Vx- 1 B. f (x) = 5x + 8; 9(x) = x-8 5 c. f (x) = 13x - 10;9(x) = *+10 3 12Exploration 9 - Finding the inverse function To find the inverse of a function, follow these steps : Example : f(x) = 3x 1. Replace the f(x) with y. 2. Switch the x and y in the equation. 3. Solve for y. 4. Replace the y with f (x) Find the inverse function of each function below: f (x) = 3x + 5 9 (x) = - 4x- 9 h(x ) =$ 4x+ 5 6 i(x) = Vx+ 7Exploration 10 Draw each function and its inverse f (x) = 2x+1 g (x ) = x - 3 * + 2 10 h(x ) = 2* 14coloration 17 cting domain The graph at right is the function f (x) = x. A. Does this graph have an inverse? B. Try to find the inverse algebraically by switching x and y. What goes wrong? C. We can find the inverse of part of the function f(x) = x by restricting the domain. What domain of f(x) = x can be used to find an inverse? Practice 12 Restrict the domain of each function so that the function is one-to-one. Then find the inverse of the domain-restricted function. A. f (x ) = 1x - 61 B. f(x) = (x+ 4)'