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Chapter 1: Problem 40

Give an example of an increasing function whose domain is the interval [0,1] but whose range does not equal the interval \([f(0), f(1)]\).

Short Answer

Expert verified
An example of an increasing function whose domain is the interval [0,1] but whose range does not equal the interval \([f(0), f(1)]\) is \( f(x) = \sqrt{x} \). This function is continuous and increasing on the interval [0,1] but does not cover the full range of values between 0 and 1, as there are no x values in [0,1] that will result in \(f(x)\) being greater than 1.

Step by step solution

01

Defining the Function

We'll use the function \( f(x) = \sqrt{x} \) as our example. Notice that \( f(x) \) is an increasing function when x is between 0 and 1.
02

Calculate the Function Values at the Endpoints of the Interval

To check the range of the function, we need to find the values of the function at the endpoints of the interval. We'll find \( f(0) \) and \( f(1) \): $$ f(0) = \sqrt{0} = 0 \\ f(1) = \sqrt{1} = 1 $$
03

Show the Range of the Function

Now we need to show that the range of function \( f(x) \) does not equal the interval \([f(0), f(1)] = [0, 1]\). It's clear that the function \( f(x) = \sqrt{x} \) is continuous and increasing on the interval [0,1]. However, for any \( \varepsilon \in (0,1) \), no value of x exists such that \( f(x) = \varepsilon^2 \) with \( \varepsilon > 1 \) and \( x \in [0,1] \). Example: Let's take \( \varepsilon = 1.1 \), then \( \varepsilon^2 = 1.21 \). No value of x in the interval [0,1] satisfies \( f(x) = 1.21\), as the maximum value of f(x) in that interval is 1, which occurs at x=1.
04

Conclusion

We provided an example of an increasing function, \( f(x) = \sqrt{x} \), whose domain is the interval [0,1]. Its range, however, does not equal the interval \([f(0), f(1)] = [0,1]\), even though it is continuous and increasing over that interval. This is because the function does not cover the full range of values between 0 and 1, as shown with \( \varepsilon = 1.1 \).

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Most popular questions from this chapter

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.

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

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