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Chapter 13: Problem 13

Determine whether each statement is true or false. If it is false, rewrite the statement so that it is true. The domain of \(f\) is the range of \(f^{-1}\)

Short Answer

Expert verified
The given statement is true: The domain of \(f\) is the range of \(f^{-1}\).

Step by step solution

01

Concept of Domain and Range of Function

Domain and Range are used to describe the set of input values (domain) and the set of possible output values (range) for a function. Domain of a function \(f\) is the set of all input values for which the function is defined, and Range of a function \(f\) is the set of all output values produced by the function for the input values in the domain.
02

Concept of Inverse Function

The inverse of a function \(f\), denoted by \(f^{-1}\), is a function that "undoes" the action of the original function \(f\) in the sense that for a function composition \(f(f^{-1}(x))=x\), and \(f^{-1}(f(x))=x\), 鈭�x in the domain of the inverse function and range of the original function respectively. In simpler terms, the inverse function reverses the process carried out by the original function.
03

Relation between Domain and Range of Inverse Function

For an inverse function, the domain and range become reversed. In other words, the domain of the inverse function is the range of the original function and the range of the inverse function is the domain of the original function.
04

True Or False Statement Evaluation

We can now check the given statement whether it's true or false. Statement: The domain of \(f\) is the range of \(f^{-1}\). Based on the concept mentioned above, the statement is true as the range of the inverse function becomes the domain of the original function i.e., range of \(f^{-1}\) is equal to domain of \(f\). Conclusion: The given statement is true, and there's no need to rewrite the statement.

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Most popular questions from this chapter

Use the formula \(A=P e^{r^{r}}\) to solve each problem. How much will Cyrus owe at the end of 6 yr if he borrows \(\$ 10,000\) at a rate of \(7.5 \%\) compounded continuously?

Determine whether each statement is true or false. If it is false, rewrite the statement so that it is true. If \(f^{-1}\) is the inverse of \(f\), then \(\left(f^{-1} \circ f\right)(x)=x\) and \(\left(f \circ f^{-1}\right)(x)=x\)

Use the formula \(A=P e^{r t}\). Raj wants to invest \(\$ 3000\) now so that it grows to \(\$ 4000\) in 4 yr. What interest rate should he look for? (Round to the nearest tenth of a percent.)

Find the inverse of each one-to-one function. $$g(x)=-4 x+8$$

Find the inverse of each one-to-one function. $$h(x)=\sqrt[3]{x-7}$$
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