Given the problem of finding a statement to explain why [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is a function, let's go through the detailed reasoning step-by-step:
1. Understanding the Given Function: The function in question is [tex]\( f(x) = 2x - 3 \)[/tex]. This is a linear function, which is a straight line when graphed.
2. Conditions for Inverse Functions: For a function to have an inverse that is also a function, it must be one-to-one. This means that each value of [tex]\( y \)[/tex] corresponds to exactly one value of [tex]\( x \)[/tex].
3. Linearity and One-to-One: Linear functions of the form [tex]\( f(x) = ax + b \)[/tex] (where [tex]\( a \neq 0 \)[/tex]) are always one-to-one. The slope [tex]\( a \)[/tex] determines the rate of change, and because it is not zero, it ensures that each [tex]\( x \)[/tex] value maps to a distinct [tex]\( y \)[/tex] value.
4. Applying the One-to-One Condition: The function [tex]\( f(x) = 2x - 3 \)[/tex] has a non-zero slope (in this case, 2). Thus, it is one-to-one, confirming that every output value [tex]\( y \)[/tex] will have exactly one corresponding input value [tex]\( x \)[/tex].
5. Conclusion Statement: Considering the above analysis, the correct statement that explains why [tex]\( f(x) = 2x - 3 \)[/tex] has an inverse relation that is a function is:
[tex]\[ \textbf{f(x) is a one-to-one function.} \][/tex]
This is the most accurate explanation:
- A one-to-one function ensures that each input maps to a unique output.
- Therefore, it has an inverse that is also a function.
Thus, the final answer is the statement:
[tex]\[ \text{f(x) is a one-to-one function.} \][/tex]
