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

Rationale

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

To find the inverse of a function, we swap the roles of x and y, then solve for y. For the function f(x) = 1/2 * x - 1, the inverse function is derived to be g(x) = 2x + 2, confirming that it undoes the operation of f.

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

This function does not represent the inverse. When substituting back into f, we would not retrieve the original x value. Instead, it only shifts the output vertically but does not reverse the effect of the original function.

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

This function also fails to be the inverse. Substituting it back into the original function f does not yield x. This function simply translates the output downwards and does not reflect the necessary algebraic operations to reverse f.

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

While this function reflects some manipulation of the original function, it does not properly reverse the operations of f. Substituting it back into f will not yield the original x value, indicating it cannot be the correct inverse.

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

Though it has the correct structure, this function still does not accurately reverse f. The constant addition of 1 fails to return to the original input of x when substituting back into the function f.

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

This function correctly represents the inverse of f. By solving f(x) = y for x and rearranging, we derive g(x) = 2x + 2, which effectively undoes the operations of f.

Conclusion

The inverse function of f(x) = 1/2 * x - 1 is found by swapping x and y, leading to g(x) = 2x + 2. The other options fail to reverse the effect of f due to incorrect transformations and constants, reinforcing the importance of proper algebraic manipulation in finding inverses.

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

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

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