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Chapter 1: Problem 68
Give an example of a function whose domain is the set of positive integers and whose range is the set of positive even integers.
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Suppose h is defined by \(h(t)=|t|+1\). What is the range of \(h\) if the domain of \(h\) is the set of negative numbers?
Give an example of a function \(f\) whose domain is the set of real numbers and such that the values of \(f(-1), f(0)\), and \(f(2)\) are given by the following table: $$ \begin{array}{r|r} {r|} {x} & {c} {f(x)} \\ \hline-1 & \sqrt{2} \\ 0 & \frac{17}{3} \\ 2 & -5 \end{array} $$
Suppose \(f\) and \(g\) are functions, each of whose domain consists of four numbers, with \(f\) and \(g\) defined by the tables below: $$ \begin{array}{c|c} {x} & {f}({x}) \\ \hline {1} & 4 \\ 2 & 5 \\ 3 & 2 \\ 4 & 3 \end{array} $$ $$ \begin{array}{c|c} x & g(x) \\ \hline 2 & 3 \\ 3 & 2 \\ 4 & 4 \\ 5 & 1 \end{array} $$ Give the table of values for \(f^{-1} \circ f\).
Give an example of a function \(f\) such that the domain of \(f\) and the range of \(f\) both equal the set of integers, but \(f\) is not a one-to-one function.
Assume that \(f\) is the function defined by $$ f(x)=\left\\{\begin{array}{ll} 2 x+9 & \text { if } x<0 \\ 3 x-10 & \text { if } x \geq 0. \end{array}\right. $$ Evaluate \(f(2)\).
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