To determine which statement explains why [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is a function, let’s analyze each option given.
1. The graph of [tex]\( f(x) \)[/tex] passes the vertical line test.
The vertical line test is used to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph does not represent a function. In this case, passing the vertical line test confirms that [tex]\( f(x) \)[/tex] is a function, but it doesn’t necessarily confirm that it has an inverse that is also a function.
2. [tex]\( f(x) \)[/tex] is a one-to-one function.
A function is one-to-one if every output is associated with exactly one input. This property is crucial because a one-to-one function will have an inverse that is also a function. This is because each input in the original function maps to a unique output, so the inverse operation will faithfully pair each output back to its unique input.
3. The graph of the inverse of [tex]\( f(x) \)[/tex] passes the horizontal line test.
The horizontal line test determines if the inverse of a function is also a function. If every horizontal line cuts the graph at most once, then the original function is one-to-one and thus its inverse is a function. However, this statement focuses on the inverse graph, so it's more indirect compared to directly establishing that the original function is one-to-one.
4. [tex]\( f(x) \)[/tex] is not a function.
This statement is incorrect because [tex]\( f(x) = 2x - 3 \)[/tex] is clearly a function.
Since the objective is to find the reason why [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse that is a function, we focus on it being one-to-one. [tex]\( f(x) = 2x - 3 \)[/tex] is a linear function with a non-zero slope (2), meaning it consistently increases or decreases, ensuring every output is unique for each input. This property ensures that [tex]\( f(x) \)[/tex] is a one-to-one function.
Therefore, the correct statement is:
[tex]\( f(x) \)[/tex] is a one-to-one function.
This ensures that [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is also a function.
