Question

The function f(x) = (x + 7)³ is one-to-one.

a. Find an equation for f1(x), the inverse function.
b. Verify that your equation is correct by showing that f(f(x)) = x and f¹(f(x)) = x.

Answer 1

a. To find the inverse function f1(x) for f(x) = (x + 7)³, we can follow these steps:

1. Replace f(x) with y: y = (x + 7)³.

2. Swap x and y: x = (y + 7)³.

3. Solve for y: y = ∛(x) - 7.

Therefore, the equation for f1(x), the inverse function, is f1(x) = ∛(x) - 7.

b. To verify that the equation is correct, we can substitute f(x) into f1(x) and f1(x) into f(x) and simplify:

1. f(f(x)) = f((x + 7)³) = ((x + 7)³ + 7)³ = (x + 7)³ + 7 = x.

2. f1(f(x)) = f1((x + 7)³) = ∛((x + 7)³) - 7 = x.

Both equations simplify to x, confirming that the equation for f1(x) is correct.