Question 41

The inverse of the function f is g(x) = 2x + 2.

The inverse function of f(x) = 1/2 * x - 1 can be found by swapping x and y and solving for y. This leads to g(x) = 2x + 2, which correctly undoes the operations of f.

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

This function does not represent the inverse of f. Applying this to f(x) would not yield x when composed, as it fails to reverse the transformation applied by f.

B) g(x) = x - 2

g(x) = x - 2 is incorrect as the inverse function. When substituting f into this function, it does not return the original input x. The operations do not align to reverse the effect of f.

C) g(x) = 2x - 1

This option is also incorrect. While it modifies the input, it does not reset to the original x when composed with f. The operations do not counteract each other as required for an inverse function.

D) g(x) = 2x + 1

This function does not serve as the inverse of f. When applied in the composition with f, it does not yield x, indicating that it doesn't reverse the operations of the original function.

E) g(x) = 2x + 2

This is the correct inverse function. By swapping x and y in f and solving for y, we arrive at g(x) = 2x + 2, effectively reversing the operations of f.

Conclusion

The correct answer, g(x) = 2x + 2, successfully inverts the function f(x) = 1/2 * x - 1, as it precisely undoes the operations of f. All other options fail to achieve this, as they do not return the original input x when composed with f, confirming their incorrectness.