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Chapter 1: Problem 65

Give an example of a function whose domain equals the set of real numbers and whose range equals the set of integers.

Short Answer

Expert verified
An example of a function whose domain equals the set of real numbers and whose range equals the set of integers is the floor function \(\lfloor x \rfloor\). This function takes any real number as input and returns the largest integer less than or equal to the input value.

Step by step solution

01

Understand the floor function

The floor function \(\lfloor x \rfloor\) is a mathematical function that, given a real number x, returns the largest integer less than or equal to x. For example, \(\lfloor 2.5 \rfloor = 2\), \(\lfloor -1.2 \rfloor = -2\), and \(\lfloor 3 \rfloor = 3\).
02

Check the domain of the floor function

The domain of a function refers to all the possible input (x) values. The floor function \(\lfloor x \rfloor\) can accept any real number as its input, so the domain is the set of all real numbers, or \(x \in \mathbb{R}\).
03

Check the range of the floor function

The range of a function represents all possible output (y) values. The floor function always returns an integer, as it rounds down to the nearest integer value. So, the range is the set of all integers, or \(y \in \mathbb{Z}\).
04

Finalize the example function with its domain and range

The floor function \(\lfloor x \rfloor\), has a domain of the set of all real numbers (\(x \in \mathbb{R}\)) and a range of the set of all integers (\(y \in \mathbb{Z}\)). Therefore, this function is an example of what the exercise asked for.

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Most popular questions from this chapter

(a) True or false: Just as every integer is either even or odd, every function whose domain is the set of integers is either an even function or an odd function. (b) Explain your answer to part (a). This means that if the answer is "true", then you should explain why every function whose domain is the set of integers is either an even function or an odd function; if the answer is "false", then you should give an example of a function whose domain is the set of integers but that is neither even nor odd.

A formula has been given defining a function \(f\) but no domain has been specified. Find the domain of each function \(f\), assuming that the domain is the set of real numbers for which the formula makes sense and produces a real number. $$ f(x)=\sqrt{|x+5|-3} $$

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

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

Give an example of two decreasing functions whose product is increasing.
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