Question 19

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

To find the inverse of the function f(x)=1/2x-1, we need to manipulate the equation to express x in terms of y. The correct inverse function is g(x)=2x+2, which successfully reverses the operation of f.

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

This function does not represent the inverse of f(x) because it does not satisfy the condition f(g(x)) = x. Substituting g(x) into f does not yield x, indicating that it is not the correct inverse.

B) g(x)=x-2

While this function may seem plausible, it fails to satisfy the inverse relationship with f(x). Evaluating f(g(x)) does not return x, which proves it is not the inverse.

C) g(x)=2x-1

This option does not meet the criteria for an inverse function. If we substitute it back into f, the result will not simplify to x, confirming it is incorrect.

D) g(x)=2x+1

This function is also incorrect as an inverse. Testing it with the original function f shows that it does not yield the identity function x, thus failing to be the inverse.

E) g(x)=2x+2

This is the correct inverse of f(x). When we substitute g(x) into f, the calculations show that f(g(x)) results in x, confirming that this function correctly reverses the operation of f.

Conclusion

The correct inverse function is g(x)=2x+2, as it satisfies the condition f(g(x)) = x. All other options fail to meet this requirement and thus cannot be considered valid inverses of the original function f.