To determine why the function [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is a function, let's analyze the provided statements step-by-step.
For a function to have an inverse that is also a function, it must be a one-to-one function. A one-to-one function is one where each [tex]\(x\)[/tex]-value maps to a unique [tex]\(y\)[/tex]-value and each [tex]\(y\)[/tex]-value maps to a unique [tex]\(x\)[/tex]-value. This property ensures that every horizontal line intersects the graph of the function at most once.
Here's why:
1. The graph of [tex]\( f(x) \)[/tex] passes the vertical line test:
- The vertical line test is used to determine if a relation is a function. If any vertical line intersects the graph of the relation at more than one point, the relation is not a function. However, this test does not determine whether the inverse is also a function.
2. [tex]\( f(x) \)[/tex] is a one-to-one function:
- This statement is crucial. For [tex]\( f(x) \)[/tex] to have an inverse that is a function, [tex]\( f(x) \)[/tex] itself must be one-to-one. A one-to-one function will pass the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once. This ensures that the inverse relation will also pass the vertical line test and thus be a function.
3. The graph of the inverse of [tex]\( f(x) \)[/tex] passes the horizontal line test:
- This statement is slightly misleading. We actually need the original function to pass the horizontal line test to ensure that its inverse will pass the vertical line test.
4. [tex]\( f(x) \)[/tex] is not a function:
- This statement is clearly incorrect because [tex]\( f(x) = 2x - 3 \)[/tex] is a linear function, which is indeed a function.
Given the linear nature of [tex]\( f(x) = 2x - 3 \)[/tex] and its continuous, non-overlapping form, we can conclude that the function [tex]\( f(x) \)[/tex] is indeed one-to-one. Therefore, among the provided options, the correct explanation is:
[tex]\( f(x) \)[/tex] is a one-to-one function.
This property of being one-to-one ensures that [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is also a function.
