The function f is defined by f(x)=1/2x-1. Which of the following functions is the inverse of f?

Rationale

g(x) = 2x + 2 is the inverse of f(x) = 1/2x - 1.

To find the inverse of a function, we swap the roles of x and y and solve for y. For the given function f(x) = 1/2x - 1, the inverse is found to be g(x) = 2x + 2, which accurately reflects the relationship defined by the original function.

A (g(x)=-1/2x+1) YOUR ANSWER

This function incorrectly applies a negative coefficient to x and does not correspond to the necessary transformations needed to reverse the original function. The slope and intercept do not align with the operations performed in f(x), making it an incorrect inverse.

B (g(x)=x-2) YOUR ANSWER

This function produces a linear transformation that does not accurately reflect the original function's behavior. Specifically, it lacks the correct slope and intercept adjustments necessary to reverse the effect of f(x), thus failing to serve as the inverse.

C (g(x)=2x-1) YOUR ANSWER

While this function has the correct slope, it does not have the proper y-intercept to be the inverse of f(x). The transformation needed to revert the original function's output is not achieved with this option, making it an incorrect choice.

D (g(x)=2x+1) YOUR ANSWER

This function maintains the correct slope but, like option C, has an incorrect y-intercept. Therefore, it does not satisfy the requirements for being the inverse of the original function.

E (g(x)=2x+2) CORRECT

This function correctly reverses f(x) by applying the appropriate operations: swapping x and y, and solving yields g(x) = 2x + 2, which accurately reflects the original relationship.

Conclusion

The inverse function is derived by switching x and y in the original equation and solving for y. Among the provided options, only g(x) = 2x + 2 effectively reverses the operation defined by f(x) = 1/2x - 1, demonstrating the principle that inverses undoes the effects of the original function.

The function f is defined by f(x)=1/2x-1. Which of the following functions is the inverse of f?

g(x) = 2x + 2 is the inverse of f(x) = 1/2x - 1.

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