Problem 8.1. For this problem, f(t) = t 1, g(t) = t 1 and h(t) = |t|. (a) Compute the multipart rules for h(f(t)) and h(g(t)) and sketch their graphs. (b) Compute the multipart rules for f(h(t)) and g(h(t)) and sketch their graphs. (c) Compute the multipart rule for h(h(t)1) and sketch the graph.

Problem 8.2. Write each of the following functions as a composition of two simpler functions: (There is more than one correct answer.) (a) y = (x 11)5. (b) y = 3 1 + x2. (c) y = 2(x 3)5 5(x 3)2 + 1 2(x 3) + 11. (d) y = 1 x2+3. (e) y =px + 1. (f) y = 2 p5 (3x 1)2. Problem 8.3. (a) Let f(x) be a linear function, f(x) = ax + b for constants a and b. Show that f(f(x)) is a linear function. (b) Find a function g(x) such that g(g(x)) = 6x 8.

Problem 8.4. Let f(x) = 1 2x + 3. (a) Sketch the graphs of f(x),f(f(x)),f(f(f(x))) on the interval 2 x 10. (b) Your graphs should all intersect at the point (6,6). The value x = 6 is called a xed point of the function f(x) since f(6) = 6; that is, 6 is xed - it doesn't move when f is applied to it. Give an explanation for why 6 is a xed point for any function f(f(f(...f(x)...))). (c) Linear functions (with the exception of f(x) = x) can have at most one xed point. Quadratic functions can have at most two. Find the xed points of the function g(x) = x2 2. (d) Give a quadratic function whose xed points are x = 2 and x = 3.

Problem 8.5. A car leaves Seattle heading east. The speed of the car in mph after m minutes is given by the function

C(m) =

70m2 10 + m2

.

(a) Find a function m = f(s) that converts seconds s into minutes m. Write out the formula for the new function C(f(s)); what does this function calculate? (b) Find a function m = g(h) that converts hours h into minutes m. Write out the formula for the new function C(g(h)); what does this function calculate? (c) Find a function z = v(s) that converts mph s into ft/sec z. Write out the formula for the new function v(C(m); what does this function calculate?

Problem 8.6. Compute the compositions f(g(x)), f(f(x)) and g(f(x)) in each case: (a) f(x) = x2,g(x) = x + 3. (b) f(x) = 1/x,g(x) = x. (c) f(x) = 9x + 2,g(x) = 1 9(x 2). (d) f(x) = 6x2 + 5,g(x) = x 4. (e) f(x) = 4x3 3,g(x) = 3 2x + 6 (f) f(x) = 2x + 1,g(x) = x3. (g) f(x) = 3,g(x) = 4x2 + 2x + 1. (h) f(x) = 4,g(x) = 0. Problem 8.7. Let y = f(z) = 4 z2 and z = g(x) = 2x + 3. Compute the composition y = f(g(x)). Find the largest possible domain of x-values so that the composition y = f(g(x)) is dened.

Problem 8.8. Suppose you have a function y = f(x) such that the domain of f(x) is 1 x 6 and the range of f(x) is 3 y 5. (a) What is the domain of f(2(x 3)) ? (b) What is the range of f(2(x 3)) ? (c) What is the domain of 2f(x) 3 ? (d) What is the range of 2f(x) 3 ? (e) Can you nd constants B and C so that the domain of f(B(x C)) is 8 x 9?

8.3. EXERCISES

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(f) Can you nd constants A and D so that the range of Af(x) + D is 0 y 1?