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Chapter 1: Problem 69

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

Short Answer

Expert verified
An example of a function whose domain is the set of positive even integers and whose range is the set of positive odd integers is \(f(n) = 2n - 1\), where n is a positive integer. This function transforms the positive even integers into positive odd integers.

Step by step solution

01

Define the function

Let's define the function as \(f(n) = 2n - 1\), where n is a positive integer.
02

Check the domain

Let's see how the function \(f(n) = 2n - 1\) behaves for the given domain (the set of positive even integers). We take the first few positive even integers and plug them into the function: - When n = 1 (even integer 2), \(f(1) = 2(1) - 1 = 1\) - When n = 2 (even integer 4), \(f(2) = 2(2) - 1 = 3\) - When n = 3 (even integer 6), \(f(3) = 2(3) - 1 = 5\) - When n = 4 (even integer 8), \(f(4) = 2(4) - 1 = 7\)
03

Check the range

As we can see from the results in step 2, the function \(f(n) = 2n - 1\) transforms the even integers (from the domain) into odd integers (forming the range). Let's check if positive odd integers {1, 3, 5, 7, 9, ...} are indeed the range of the function. - For n = 1, \(2n - 1 = 1\) - For n = 2, \(2n - 1 = 3\) - For n = 3, \(2n - 1 = 5\) - For n = 4, \(2n - 1 = 7\) As we can see, for each positive integer n, the function yields a positive odd integer, and therefore, the range of the function is indeed the set of positive odd integers. So, the function \(f(n) = 2n - 1\) is an example of a function, which has its domain as the set of positive even integers and its range as the set of positive odd integers.

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Most popular questions from this chapter

For Exercises 51-56, suppose h is defined by \(h(t)=|t|+1\). What is the range of \(h\) if the domain of \(h\) is the interval (1,4]\(?\)

Suppose the only information you know about a function \(f\) is that the domain of \(f\) is the set of real numbers and that \(f(1)=1, f(2)=4\), \(f(3)=9,\) and \(f(4)=16 .\) What can you say about the value of \(f(5) ?\) [Hint: The answer to this problem is not "25". The shortest correct answer is just one word.]

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 \(I(x)=x\). $$ f(x)=\frac{1}{3 x+2} $$

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 \(I(x)=x\). $$ f(x)=\frac{4}{5 x-3} $$

For Exercises \(33-40,\) assume that \(f\) is the function defined by $$ f(x)=\left\\{\begin{array}{ll} 2 x+9 & \text { if } x<0 \\ 3 x-10 & \text { if } x \geq 0 \end{array}\right. $$ Evaluate \(f(1)\).
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