Question

The equation defines a one-to-one function f. f(x)=(x+8)^(3) Determine f^(-1). f^(-1)(x)

Answer 1

To find the inverse function f^(-1)(x) of the function f(x) = (x+8)^3, we swap x and y and then solve for y, giving us the inverse function f^(-1)(x) = ∛(x) - 8, where ∛ denotes the cube root.

Step-by-step explanation:

The student is asking how to find the inverse function f^(-1)(x) of the one-to-one function f(x) = (x+8)^3. To find the inverse function, we need to switch the roles of x and y and then solve for y:

Let y = f(x) = (x+8)^3. To find the inverse, we swap x and y to get x = (y+8)^3. We then solve this equation for y:

1. Take the cube root of both sides: y + 8 = ∛(x).
2. Subtract 8 from both sides to isolate y: y = ∛(x) - 8.

Therefore, the inverse function is f^(-1)(x) = ∛(x) - 8, where ∛ represents the cube root function.