The correct answer is:

Yes, each element in the range has only one domain value.

Explanation:

For $f(x)$ to have an inverse that is also a function, each element in the range (output values) must correspond to exactly one element in the domain (input values).

Given the function $f(x) = \{(1, 4), (2, 3), (5, 8), (4, 7)\}$:

• The range (outputs) is $\{4, 3, 8, 7\}$.
• The domain (inputs) is $\{1, 2, 5, 4\}$.

When checking if $f^{-1}(x)$ is a function, we look at the elements in the range of $f(x)$ to ensure each corresponds to exactly one element in the domain. Since this is true for all the pairs, $f^{-1}(x)$ is indeed a function.

Relative Questions:

1. What is the general criterion for a function to have an inverse that is also a function?
2. How do you find the inverse of a function given a set of ordered pairs?
3. If the function $f(x)$ were given by $f(x) = \{(1, 4), (2, 4), (5, 8), (4, 7)\}$, would $f^{-1}(x)$ be a function? Why or why not?
4. What would the inverse $f^{-1}(x)$ look like for the given set of ordered pairs?
5. Can a function have more than one inverse? Why or why not?

Tip:

When determining if the inverse of a function is also a function, always check that each output value in the original function corresponds to exactly one input value.