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Chapter 1: Problem 73

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

Give an example of a one-to-one function whose domain equals the set of integers and whose range equals the set of positive integers.

Suppose $$h(x)=\left(\frac{x^{2}+1}{x-1}-1\right)^{3}$$ (a) If \(f(x)=x^{3},\) then find a function \(g\) such that \(h=f \circ g\) (b) If \(f(x)=(x-1)^{3}\), then find a function \(g\) such that \(h=f \circ g\).

Suppose \(g(x)=x^{2}+4\), with the domain of \(g\) being the set of positive numbers. Evaluate \(g^{-1}(7)\).

Suppose \(h(x)=3 x^{2}-4,\) where the domain of \(h\) is the set of positive numbers. Find a formula for \(h^{-1}\).
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