Final answer:
The correct expression for the inverse function f^(-1)(x) of f(x) = 1/9x - 2 is f^(-1)(x) = 9x + 2, which corresponds to option c).
Step-by-step explanation:
To find the inverse function f^(-1)(x) of f(x) = 1/9x - 2, begin by setting f(x) equal to y. Then, interchange x and y to solve for y in terms of x. The original function equation is y = 1/9x - 2.
Swap x and y to get x = 1/9y - 2. To isolate y, add 2 to both sides: x + 2 = 1/9y. Multiply both sides by 9 to solve for y, obtaining 9x + 18 = y, or y = 9x + 18. Simplifying by subtracting 18 from both sides yields the inverse function f^(-1)(x) = 9x + 2.
The process of finding an inverse function involves interchanging x and y in the original function equation, solving for y, and then re-expressing the equation solved for y in terms of x to obtain the inverse function. In this case, f^(-1)(x) = 9x + 2 reflects the correct inverse function of f(x) = 1/9x - 2, aligning with option c). Therefore, the accurate representation of the inverse function is f^(-1)(x) = 9x + 2 among the provided options.