Question

f(x) = {(1, 4), (2, 3), (5, 8), (4, 7)} is f⁻¹(x) a function and why?
no, there are only a limited number of ordered pairs and can’t be determined.
yes, each element in the range has only one domain value.
yes, each element in the domain has only one range value.
no, we don’t know the original function and therefore can’t make the determination.

Explanation:

To determine if the inverse of a function \( f(x) \) is a function, we use the horizontal line test for the original function (or check if the inverse relation is a function by verifying that each element in the range of \( f(x) \) maps to only one element in the domain).

First, find the inverse relation of \( f(x)=\{(1, 4), (2, 3), (5, 8), (4, 7)\} \). The inverse relation \( f^{-1}(x) \) is obtained by swapping the \( x \)- and \( y \)-values of each ordered pair, so \( f^{-1}(x)=\{(4, 1), (3, 2), (8, 5), (7, 4)\} \).

Now, check if \( f^{-1}(x) \) is a function. A relation is a function if each input (element in the domain) has exactly one output (element in the range). For \( f^{-1}(x) \), the domain elements are 4, 3, 8, 7. Each of these domain elements maps to exactly one range element (1, 2, 5, 4 respectively). In other words, each element in the range of \( f(x) \) (which is the domain of \( f^{-1}(x) \)) has only one domain value of \( f(x) \) (which is the range value of \( f^{-1}(x) \)).

Let's analyze the options:

• The first option is incorrect because the number of ordered pairs doesn't prevent us from determining if it's a function.
• The second option is correct: Each element in the range of \( f(x) \) (domain of \( f^{-1}(x) \)) has only one domain value of \( f(x) \) (range of \( f^{-1}(x) \)), so \( f^{-1}(x) \) is a function.
• The third option describes the original function \( f(x) \) being a function (each domain element has one range value), but we need to check the inverse, so this is not the reason for \( f^{-1}(x) \) being a function.
• The fourth option is incorrect because we do know the original function (given as a set of ordered pairs) and can determine the inverse relation.

Answer:

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