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Chapter 1: Problem 70

Give an example of a function whose domain is the set of integers and whose range is the set of positive integers.

Short Answer

Expert verified
An example of a function whose domain is the set of integers and whose range is the set of positive integers is \(f(x) = x^2 + 1\), where \(x\) is an integer. This function takes integer inputs and always produces positive integer outputs as required.

Step by step solution

01

Identify a suitable function

After analyzing the requirements, a suitable function is \(f(x) = x^2 + 1\), where \(x\) is an integer. This function will take integer inputs (x) and always produce positive integer outputs.
02

Verify the output as a positive integer

To ensure that the outputs are always positive integers, let's check the function for a few integer inputs: 1) When \(x = -2\), we get: \(f(-2) = (-2)^2 + 1 = 4 + 1 = 5\) 2) When \(x = 0\), we get: \(f(0) = (0)^2 + 1 = 0 + 1 = 1\) 3) When \(x = 2\), we get: \(f(2) = (2)^2 + 1 = 4 + 1 = 5\) These examples prove that the function \(f(x) = x^2 + 1\) satisfies the requirements of domain and range. It takes integers as input and produces positive integers as output.

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Most popular questions from this chapter

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 \(f^{-1} \circ g^{-1}\).

Suppose $$ h(x)=2+\sqrt{\frac{1}{x^{2}+1}} $$ (a) If \(g(x)=\frac{1}{x^{2}+1},\) then find a function \(f\) such that \(\bar{h}=f \circ g\) (b) If \(g(x)=x^{2},\) then find a function \(f\) such that \(h=f \circ g\).

Suppose \(f\) is a function whose domain equals \\{2,4,7,8,9\\} and whose range equals \(\\{-3,0,2,6,7\\} .\) Explain why \(f\) is a one-to-one function.

A constant function is a function whose value is the same at every number in its domain. For example, the function \(f\) defined by \(f(x)=4\) for every number \(x\) is a constant function. Give an example of three functions \(f, g,\) and \(h\), none of which is a constant function, such that \(f \circ h=g \circ h\) but \(f\) is not equal to \(g\).

In Exercises \(37-40,\) find functions \(f\) and \(g,\) each simpler than the given function \(h\), such that \(\boldsymbol{h}=\boldsymbol{f} \circ \mathrm{g}\) $$ h(x)=\frac{3}{2+x^{2}} $$
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