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Chapter 1: Problem 69

Give an example of a function whose domain is the interval [0,1] and whose range is the interval (0,1) .

Short Answer

Expert verified
The example of a function whose domain is the interval [0,1] and whose range is the interval (0,1) is: \(f(x) = 1 - e^{-x}\)

Step by step solution

01

Understanding Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this case, we want a function with a domain of [0,1] and a range of (0,1). This means that the function should be able to take any value from the interval [0,1] and produce a corresponding output value from the interval (0,1).
02

Choose an Appropriate Function

One example of such a function is: \(f(x) = 1 - e^{cx}\), where c is a constant. Let's proceed to find a suitable value for the constant 'c' in order to have the required domain and range.
03

Obtain the Interval for the Domain

We will start by finding the value of the function at each endpoint of the domain [0,1]: For the left endpoint 0, we have: \(f(0) = 1 - e^{c\cdot 0} = 1 - e^0 = 1 - 1 = 0\) For the right endpoint 1, we have: \(f(1) = 1 - e^{c\cdot 1} = 1 - e^c\) In this case, \(e^c\) must lie in the interval (0,1) so that \(f(1)\) also lies in the interval (0,1).
04

Choose a Suitable Constant

Let's choose \(c = -1\). This makes the function as \(f(x) = 1 - e^{-x}\). Verify that the chosen constant ensures the desired domain and range: For \(x = 0\): \(f(0) = 1 - e^0 = 1 - 1 = 0\) For \(x = 1\): \(f(1) = 1 - e^{-1} = 1 - \frac{1}{e}\) Since \(0 < \frac{1}{e} < 1\), the range of the function lies in the interval (0,1).
05

Final Answer

The example of a function whose domain is the interval [0,1] and whose range is the interval (0,1) is: \(f(x) = 1 - e^{-x}\)

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Most popular questions from this chapter

Suppose \(g\) is the function whose domain is the interval [-2,2] , with \(g\) defined on this domain by the formula $$g(x)=\left(5 x^{2}+3\right)^{7777}$$ Explain why \(g\) is not a one-to-one function.

Explain why an even function whose domain contains a nonzero number cannot be a one-to-one function.

Suppose \(f\) is a function and a function \(g\) is defined by the given expression. (a) Write \(g\) as the composition of \(f\) and one or two linear functions. (b) Describe how the graph of \(g\) is obtained from the graph of \(f\). \( g(x)=3 f(x)-2\)

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

Suppose \(f\) and \(g\) are functions, each with domain of four numbers, with \(f\) and \(g\) defined by the tables below: $$\begin{array}{c|c}x & f(x) \\\\\hline 1 & 4 \\\2 & 5 \\\3 & 2 \\\4 & 3\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & 3 \\\3 & 2 \\\4 & 4 \\\5 & 1\end{array}$$ Sketch the graph of \(g\).
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