Problem 94 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The domain of \(f\) is the same as the range of \(f^{-1}\).

Step-by-Step Solution

Answer

The statement 'The domain of \(f\) is the same as the range of \(f^{-1}\)' is True.

Step 1: Understanding the Definition

To begin, it's important to know what is meant by 'domain' and 'range'. The 'domain' of a function is the complete set of possible values of the independent variable. In the case of function \(f\), it is all the values that could be input into 'f'. On the other hand, the 'range' of a function is the complete set of possible output values, which is the outcome from the function for all the domain.

Step 2: Analyzing the Statement

The statement says that 'The domain of \(f\) is the same as the range of \(f^{-1}\)'. Now, the symbol \(f^{-1}\) stands for the inverse function of \(f\), which 'undoes' the effect of \(f\). When it comes to functions and their inverses, if \(f(a) = b\), then \(f^{-1}(b) = a\), almost like exchanging input and output. This means that, typically, the domain of \(f\) is the range of \(f^{-1}\), and vice versa.

Step 3: Evaluating the Statement

Given the definition of the inverse function and its relationship with \(f\), it's clear that the statement 'The domain of \(f\) is the same as the range of \(f^{-1}\)' is indeed True. No modification is needed.