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Chapter 1: Problem 63

Give an example of two different functions \(f\) and \(g\), both of which have the set of real numbers as their domain, such that \(f(x)=g(x)\) for every rational number \(x\).

Short Answer

Expert verified
The two functions that satisfy the given condition are: 1. \(f(x) = x\) 2. \(g(x) = \begin{cases} x & \text{if } x \text{ is rational} \\ -x & \text{if } x \text{ is irrational} \\ \end{cases} \) These functions have the same values for every rational number x, and different values for irrational numbers.

Step by step solution

01

Choose function f

We can pick a simple function for f. Let's use the identity function: \(f(x) = x\).
02

Define function g

We need to find another function, g, that has the same values as f for every rational number, meaning \(g(x)=x\) for every rational number x, but is still different from f. Consider the piecewise function g: \[g(x) = \begin{cases} x & \text{if } x \text{ is rational} \\ -x & \text{if } x \text{ is irrational} \\ \end{cases} \]
03

Comparing functions f and g

Now, let's compare these two functions, f and g, to verify if they satisfy the condition given: 1. For any rational number x, f(x) = x and g(x) = x, so f(x) = g(x) for every rational number x, which is required according to the exercice. 2. For irrational number x, f(x) = x and g(x) = -x. We can see that the functions are different for irrational numbers. Thus, we have found two functions that satisfy the given condition: f(x) = x and g(x) is a piecewise function defined by the above rules.

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Most popular questions from this chapter

For each of the functions \(f\) given in Exercises \(13-\) 22: (a) Find the domain of \(f\). (b) Find the range of \(f\). (c) Find a formula for \(f^{-1}\). (d) Find the domain of \(\boldsymbol{f}^{-1}\). (e) Find the range of \(f^{-1}\). You can check your solutions to part (c) by verify. ing that \(f^{-1} \circ f=I\) and \(f \circ f^{-1}=I\) (recall that \(I\) is the function defined by \(I(x)=x\). $$ \begin{aligned} &f(x)=x^{2}+8, \text { where the domain of } f \text { equals }\\\ &(0, \infty) \end{aligned} $$

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 \(f\) is a one-to-one function. Explain why the inverse of the inverse of \(f\) equals \(f . \underline{\text { In }}\) other words, explain why $$ \left(f^{-1}\right)^{-1}=f $$

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

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