Which statement could be used to explain why $f(x)=2 x-3$ has an inverse relation that is a function?
A. The graph of $f(x)$ passes the vertical line test.
B. $f(x)$ is a one-to-one function.
C. The graph of the inverse of $f(x)$ passes the horizontal line test.
D. $f(x)$ is not a function.
Answer (2)
The vertical line test confirms f ( x ) is a function, but not if its inverse is a function.
f ( x ) being one-to-one guarantees its inverse is also a function.
The horizontal line test on the inverse is equivalent to the vertical line test on the original function, which is not relevant.
f ( x ) is indeed a function, so the last option is incorrect. The answer is that f ( x ) is a one-to-one function. f ( x ) is a one-to-one function.
Explanation
Analyzing the Problem The question asks which statement explains why the function f ( x ) = 2 x − 3 has an inverse relation that is also a function. Let's analyze each option.
Vertical Line Test The first option states 'The graph of f ( x ) passes the vertical line test.' The vertical line test determines if a relation is a function. Since f ( x ) is already given as a function, this statement just confirms that f ( x ) is a function, which we already know. It doesn't tell us anything about its inverse.
One-to-one Function The second option states ' f ( x ) is a one-to-one function.' A function has an inverse that is also a function if and only if the original function is one-to-one. This means that for every y value, there is only one x value. This is the correct condition for the inverse to be a function.
Horizontal Line Test The third option states 'The graph of the inverse of f ( x ) passes the horizontal line test.' If the inverse of f ( x ) passes the horizontal line test, it means the original function f ( x ) passes the vertical line test, which we already established. This doesn't explain why the inverse is a function.
Function Check The fourth option states ' f ( x ) is not a function.' This is false because f ( x ) = 2 x − 3 is a linear function, and all linear functions are functions.
Conclusion Therefore, the correct statement is ' f ( x ) is a one-to-one function.' This is because a function has an inverse that is also a function if and only if the original function is one-to-one.
Examples
Imagine you have a machine that converts temperatures from Celsius to Fahrenheit using the function f ( x ) = 5 9 x + 32 . If this machine has an inverse function, it means you can convert Fahrenheit back to Celsius without any ambiguity. The 'one-to-one' property ensures that each Fahrenheit temperature corresponds to only one Celsius temperature, allowing for a perfect reverse conversion. This principle applies to many real-world conversions and encoding processes where reversibility is crucial.
The correct explanation for why f ( x ) = 2 x − 3 has an inverse that is also a function is that it is a one-to-one function (Option B). This property guarantees that each output corresponds to a unique input, allowing the inverse to exist as a function. Other options either confirm basic aspects of functions or are incorrect about f ( x ) .
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