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Chapter 1: Problem 106
Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(g^{-1} \circ f^{-1}\right)(-3)$$
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Determine whether the function is even, odd, or neither (a) algebraically, (b) graphically by using a graphing utility to graph the function, and (c) numerically by using the table feature of the graphing utility to compare \(f(x)\) and \(f(-x)\) for several values of \(x\). $$g(x)=x^{3}-5 x$$
Identify the terms. Then identify the coefficients of the variable terms of the expression. $$\frac{x}{3}-5 x^{2}+x^{3}$$
The function in Example 9 can be decomposed in other ways. For which of the following pairs of functions is \(h(x)=\frac{1}{(x-2)^{2}}\) equal to \(f(g(x)) ?\) (a) \(g(x)=\frac{1}{x-2}\) and \(f(x)=x^{2}\) (b) \(g(x)=x^{2}\) and \(f(x)=\frac{1}{x-2}\) (c) \(g(x)=(x-2)^{2}\) and \(f(x)=\frac{1}{x}\)
You can encode and decode messages using functions and their inverses. To code a message, first translate the letters to numbers using 1 for "A," 2 for "B," and so on. Use 0 for a space. So, "A ball" becomes 1 0 2 1 12 12. Then, use a one-to-one function to convert to coded numbers. Using \(f(x)=2 x-1,\) "A ball" becomes 1 ?1 3 1 23 23. (a) Encode "Call me later" using the function \(f(x)=5 x+4.\) (b) Find the inverse function of \(f(x)=5 x+4\) and use it to decode 119 44 9 104 4 104 49 69 29.
Use examples to hypothesize whether the product of an odd function and an even function is even or odd. Then prove your hypothesis.
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