x 4 D: (4, ); g( f (x)) = 1

Transcription

1 Honors Math 4 Describing Functions One Giant Review Name Answer Key 1. Let f (x) = x, g(x) = 6x 3, h(x) = x 3 a. f (g(h(x))) = 2x 3 b. h( f (g(x))) = 1 3 6x 3 c. f ( f ( f (x))) = x Let f (x) = x 3, g(x) = x, h(x) = x 4, j(x) = 2x. Express each function k as a composite of three of these four functions. a. k(x) = 2(x 4) 3 = j( f (h(x))) b. k(x) = (2x 8) 3 = f ( j(h(x))) c. k(x) = x 3 4 = g(h( f (x))) 3. Find rules for (f o g)(x) and (g o f)(x) and give the domain of each composite function. a. f (x) = 2x, g(x) = 16 x 2 f (g(x)) = 2 16 x 2 D: [ 4, 4]; g( f (x)) = 2 4 x 2 D: [ 2, 2] b. f (x) = x, g(x) = 1 1 f (g(x)) = x 4 x 4 D: (4, ); g( f (x)) = 1 D: [0, 16) (16, ) x 4 4. If f (x) = 2x 1, show that f ( f (x)) = 4x 3. Find f ( f ( f (x))). Has the domain changed? Has the range? f(f(f(x))) = 8x 7 Composing linear functions gives linear functions D: x R, R: y R 5. Suppose a function f has an inverse. If f(2) = 6 and f(3) = 7, find: a. f 1 (6) = 2 b. f 1 f (3) ( ) = 3 ( ) = 7 c. f f 1 (7) 6. Sketch g(x) = x 3 + x 2 and explain why it is not invertible. See your calculator for the graph. Between 1 and 0 the function will not pass the horizontal line test (the function is not one-to-one). 7. For each of the functions below, find the domain and range, sketch the function, and determine if it is invertible. If it is invertible, add the inverse to your sketch and find it algebraically. a. f (x) = x 2 D: x R, R: y 2 not invertible (it s a V) b. g(q) = mq + z, where m 0 D: g R, R: q R q(g) = 1 m (g z) (all lines are invertible) c. c(b) = 1 D: b 2 0, R: c > 0 (not invertible, it s a volcano) b d. s(t) = 5 t D: t 5, R: s 0 t(s) = 5 s 2 e. s(t) = 3 5 t 3 D: t R, R: s R t(s) = 3 5 s 3 f. h(k) = (k 1) 2 +1, k 1 D: given, R: h 1 k(h) = h If f is a linear function such that f (x + 2) f (x) = 6, find the value of f 1 (x + 2) f 1 (x).

2 Describing Functions Review page 2 of 6 m f = f (x + 2) f (x) (x + 2) (x) = 6 2 so m f 1 = (x + 2) (x) f (x + 2) f (x) = 2 6 = f 1 (x + 6) f 1 (x) (x + 6) (x) so f 1 (x + 2) f 1 (x) = Suppose a, b, and c are constants such that a 0. Let P(x) = ax 2 + bx + c for x b. Find a rule for 2a P 1 (x). P 1 (x) = b 2a b2 4a(c x) 2a 10. Here are two functions that you ve seen on a previous assignment: E = f(s) and C = g(e). a. Is the inverse of f also a function? If yes, make a table of values and a graph for the inverse function f 1. If no, explain why not. E = f(s) is not one-to-one (fails the horizontal line test), so it is not invertible b. Is the inverse of g also a function? If yes, make a table of values and a graph for the inverse function g 1. If no, explain why not. See the table at the right. 11. A function is said to be invertible if the function s inverse is also a function. A function is invertible if and only if the function is one-to-one (only one x for each y, or visually, passes the horizontal line test). a. Which of the 12 basic functions (from section 1.3) are invertible? For each invertible function, sketch a graph of its inverse. x, 1 x, ex, ln x b. For which of the basic functions will the inverse also be one of the basic functions? Same 4. c. We discussed during class that the x 2 function is not invertible, but x 2 restricted to x 0 is invertible. Find another basic function f(x) that is not invertible, but can be made invertible through a restriction on x. Sketch the graph of the restricted f(x) and its inverse. x, x 0 C g 1 (C) = E A graph of v(x) is given on the grid below. a. Is v(x) an invertible function? Justify your answer. No, not one-to-one b. Draw the inverse of v(x) anyway. Explain why it isn t a function. Fails vertical line test c. Find a restriction of the domain of v(x) so that the restricted graph will be invertible. x 1 d. Using your restriction from part c, graph the inverse function v 1 (x). Top branch e. Find a second possible answer to part c. x 1, bottom branch

3 Describing Functions Review page 3 of a. Use the following information to sketch f(x). b. Sketch f 1 (x). In class c. Find the following: Domain of f 1 (x): (, 4) ( 4, ) Range of f 1 (x): (, 3) ( 3, ) x f 1 (x) = f 1 (x) = 0 x x 4 f 1 (x) = 3 x 4 + f 1 (x) = x 0 f 1 (x) = 0 x 0 + f 1 (x) = 3 d. Compare the limits of f(x) and f 1 (x). What do you notice? All the vertical and horizontal limits switch. 14. Do the following for the f(x) parabola and g(x) the wiggly curve given below. a. Draw the graph of f + g (on the left). b. Draw the graph of f g (on the right). c. Draw the graph of the average of the functions, (f(x) + g(x)) on the right axes below. How is it related to one of the other graphs that you ve drawn? Half of f + g. d. When you draw the average (f(x) + g(x)) on the same axes as f(x) and g(x), where is the average graph located, in relation to the original functions? Runs halfway between the two curves.

4 Describing Functions Review page 4 of 6 Proofs involving even and odd functions 15. State whether the functions are odd, even, or neither. Support graphically and confirm algebraically 3 a. g( x) = 1+ ( x) = = g(x) so Even 1+ x b. f ( x) = ( x) ( x) = x x f (x) or f (x) so Neither c. h( x) = 1 x = 1 x = h(x) so Odd 16. Determine whether each of the following is even, odd, or neither then prove it. a. The sum of two odd functions. f ( x) + g( x) = f (x) g(x) = f (x) + g(x) b. The sum of an even and an odd function. Let f(x) be the odd function and g(x) be the even function. f ( x) + g( x) = f (x) + g(x) f (x) + g(x) c. The product of two even functions. f ( x) + g( x) = f (x) + g(x) so the product is Even d. The product of an even function and an odd function. f ( x) g( x) = f (x) g(x) = f (x) g(x) e. The quotient of two odd functions. f ( x) g( x) = f (x) g(x) = f (x) g(x) ( ) so the sum of two odd functions is Odd ( ) or ( f (x) + g(x) ) so the sum is Neither ( ) so the product is Odd so the quotient is Even 17. Based on your answers from question 16, see how many conjectures (educated guesses) you can make about results involving even and odd functions. You do not need to prove your results (but you may want to in order to determine if you were right). This problem is all you. 18. First try this example. Let f(x) = 2x 3 + 3x 2 + 4x + 5. Note: You can prove that f(x) is neither even nor odd. a. Calculate the function formula for E(x). Simplify your answer as much as possible. Then algebraically prove that this E(x) is an even function. E(x) = 3x 2 + 5, an even function b. Calculate the function formula for O(x). Simplify your answer as much as possible. Then algebraically prove that this O(x) is an odd function. O(x) = 2x 3 + 4x, an odd function

5 Describing Functions Review page 5 of Now let f(x) stand for any function. a. Prove that E(x) = (f(x) + f( x)) is an even function. E( x) = (f( x) + f( ( x))) ) = (f( x) + f(x)) = (f(x) + f( x)) = E(x) b. Prove that O(x) = (f(x) f( x)) is an odd function. O( x) = (f( x) f( ( x))) ) = (f( x) f(x)) = (f(x) f( x)) = O(x) c. Prove that f(x) = E(x) + O(x). E(x) + O(x) = (f(x) + f( x)) + (f(x) f( x)) = [f(x) + f(x) + f( x) f( x)] = f(x) To summarize: What you ve just proved is that any function can be decomposed into an even part and an odd part. That s pretty interesting. 20. Given below are graphs of a function f(x) along with its f( x). Draw graphs of E(x) and O(x). They should end up appearing even and odd respectively. a. Draw E(x) = (f(x) + f( x)). b. Draw O(x) = (f(x) f( x)). You may use your calculator to help generate the graphs in the following problem, but remember that calculators don t always do a good job at accurately representing asymptotes. 21. Consider the function f(x) =. a. Graph f(x). b. Find a function formula for f( x), then graph f( x). How are the graphs of f(x) and f( x) related to each other? f( x) =, it is f(x) flipped over the y-axis. c. Find a function formula for E(x) = (f(x) + f( x)), then graph E(x). E(x) = (f(x) + f( x)) =

6 Describing Functions Review page 6 of 6 d. Find a function formula for O(x) = (f(x) f( x)), then graph O(x). O(x) = (f(x) f( x)) = e. Compare the locations of the discontinuities (vertical asymptotes) for the four graphs f(x), f( x), E(x), and O(x). 22. a. The inverse of an even function is always/sometimes/never a function. Explain. Even functions have symmetry over the y-axis so they never pass the horizontal line test. The inverse is never a function. b. The inverse of an odd function is always/sometimes/never a function. Explain. Odd functions have symmetry around the origin so they are always one-to-one. The inverse is always a function.