Question

Consider the function f(x) = x⁴ + 4, where x ≥ 0. Find an explicit formula for f⁻¹.

a) f⁻¹(x) = ∛(x - 4)
b) f⁻¹(x) = ∜(x) - 4
c) f⁻¹(x) = ∛(x + 4)
d) f⁻¹(x) = ∛(x) + 4

Answer 1

The explicit formula for the inverse function f⁻¹(x) of f(x) = x⁴ + 4, where x ≥ 0, is found by solving for x in terms of y and then interchanging x and y, which gives us f⁻¹(x) = √⁴(x - 4), corresponding to option (a).

Step-by-step explanation:

To find the explicit formula for the inverse function f⁻¹(x) of the given function f(x) = x⁴ + 4, where x ≥ 0, we must solve for x in terms of y, such that y = x⁴ + 4. Since we're looking for f⁻¹(x), we interchange x and y to get x = y⁴ + 4.

Now we solve for y:
x - 4 = y⁴
√⁴(x - 4) = y

Given that x - 4 must be non-negative because x ≥ 4 (ensured by the domain x ≥ 0 and the nature of the original function), we can apply the fourth root to both sides, leading to:

f⁻¹(x) = √⁴(x - 4)

This matches option (a). Therefore, the explicit formula for the inverse function f⁻¹(x) is:

f⁻¹(x) = √⁴(x - 4)