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Chapter 1: Problem 68

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

For each of the functions \(f\) given. (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(f^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part (c) by verifying that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I\) (recall that \(I\) is the function defined by \(I(x)=x\) ). \(f(x)=2 x^{2}+5,\) where the domain of \(f\) equals \((0, \infty)\).

Suppose you are exchanging cur. rency in the London airport. The currency exchange service there only makes transactions in which one of the two currencies is British pounds, but you want to exchange dollars for Euros. Thus you first need to exchange dollars for British pounds, then exchange British pounds for Euros. At the time you want to make the exchange, the function \(f\) for exchanging dollars for British pounds is given by the formula \(f(d)=0.66 d-1\) and the function \(g\) for exchanging British pounds for Euros is given by the formula \(g(p)=1.23 p-2\) The subtraction of 1 or 2 in the number of British pounds or Euros that you receive is the fee charged by the currency exchange service for each transaction. Find a formula for the function given by your answer to Problem \(55 .\)

Suppose \(g(x)=x^{7}+x^{3} .\) Evaluate $$\left(g^{-1}(4)\right)^{7}+\left(g^{-1}(4)\right)^{3}+1$$

Show that the product of two even functions (with the same domain) is an even function.

Suppose the income tax function in Example 2 of Section 1.1 is changed so that $$g(x)=0.15 x-450 \quad \text { if } 8500
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