Model 2: The Identity Function and Inverses There is a special function called the identity function, which we denote by id. This function always outputs exactly what was input: id 12. Evaluate the following. (a) id(0) = (b) id(1) =. (c) id(3) = _ (d) id(-2) = 13. Draw the graph of y = id(x) on the axes below. . . . ..... . . ...... - . . . . . .. N - . . . . ... ...... ....... f..... . wf...... -4 -3 -2 -1 2 ....... . . . . .. ....; .... .. . . . . .J . . . . ..." -2 . . . ... . .. .... . .... .. - 14. Suppose we have a function f that adds one to its input. We need to create a function g that is an "undo" function for f. What should g do? 15. Fill in the diagrams below to compute formulas for (go f)(x) and (fog)(x). (go f ) (fog) f x+1 g g f 16. Rewrite the results of #15 below. (go f) (fog) 5 5= id and 1 - - . ' t] t 0 f ) DGniti'm: SUPPOSE f is a function. Then the function f 1 Is the inverse of f DIOVIded m (f (f-1 o f) = id for all values in the functions' domains. 1?. Verify that the function 9 you chose in #14 is in fact 1' \". Revise your miswer if needed. 18. For each function f below, identify the formula for its inverse function f']. (a) I+'> :c5 r)f\"--> 19. Suppose f(:z:) = $2 and 9(3) = J}. (a) Recall from #12 that id(2) = . Compute (gof)(2). (b) Based on part (a), does (9 of) = id? __ f (c) Based on part (b), is g the inverse of f? (d) Suppose the domain of f is restricted to [0,00). On this domain, does (9 o f) = id? Explain your answer with a complete sentence. (e) Suppose the domain of f is instead restricted to (oo,0]. How could you modify the formula for 9(1') so that g is the inverse of f on this domain? F hen taking the square root of both sides of an equation. We are inverting a . . 19This problem illustrates why we need to include plus or minus \\\\' squaring function, but we don't know which inverse to use! as Model 3: Properties of the Inverse 20. Suppose f is a one-to-one function with inverse f-1. Complete the following. (a) Suppose f(a) = b. Represent this in the diagram: f (b) Use the fact that (f-1 of) = id to add to the diagram: id f -1 (c) According to the diagram in part (b), f-1(-) (d) So when the point (a, b) is on the graph of y = f(x), it means the point (-,-) is on the graph of y = f-1(x). 21. The graph of y = g(x) is given below. On the same axes, draw the graph of y = 9-1(x). Hint: Start by plotting the endpoints and the corner. . . . ... 1 . . .. . ............. { . ....... 4+. . . .. .. CO .. .;.. .. . ...].........1"; .... .... -...:60 -... .. . .. ...... ...."." of ... -4 -3 -2 -1 .. NO .......' . 3. ....... . . . .. . . ........ -2 - ..!... -3 ..... ...!.. . ! . .. . . . ... -4 .. ... . ..... -5 - 22. Draw (as a dashed line) the line of symmetry between the graphs of g and g- above. This line of symmetry is the same as the graph of what special function? Hint: #13 Recall: A function is one-to-one when each of its outputs comes from only one input. The graph of a one-to-one function will pass the horizontal line test.. f . Therefore, 23. The graph of f\" is obtained by swapping the coordinates of each pomt on the graph 0 f - the domain of r1 is equivalent to the ______.__ of f, and o the range of f'1 is equivalent to the .~ Of 1" 25c + 3 24. We can also nd the formula. for f '1 by swapping coordinates. Suppose f (1 ) = 7:1: - 5' (a) Start with y = g: + g. In this equation, y represents f (cc). -1 . , . t' n, represents f (33)- (b) Swap coordinates: replace the y with r, and replace all x's With y s. In this new equa 10 y (c) Now we need to solve for y in the new equation. Start by multiplying both sides by the denominator. (d) After distributing, you should have two terms on each side. Arrange the equatiou so that terms with a factor of y are on one side and terms without a factor of y are on the other side. (e) Factor out the y, then divide both sides by the other factor on that side. N ow y should be isolated, and you have the formula for f\"(z). 1:9 Practice: Composition of Functions We 1. Let g(x) = 12 + 1 and let h(x) = Vi + 6. (a) Evaluate (go h)(3). (b) Find (goh)(x). (c) State the domain of (go h)(z) using interval notation. (d) Find (hog)(x). (e) State the domain of (hog)(x) using interval notation. 2. Let f(x) = _ x+6 x + 3 and let g(x) = 2x + 3. (a) Find (fog)(r). (b) State the domain of (fog)(r) using interval notation. 505. Cc Practice: Inverse Functions 3. Let f(x) = V3x - 8. Find f-1(x). (0) : 4. Let g(r) = Vx - 4+7. (a) Find g-(x). (b) State the domain of g"(x) using interval notation. 5. Let h(I) = 71 - 8 (a) Find h-(I). (b) State the domain and range of h-(r) using interval notation. 60