Question

Find the inverse of the function f(x) = 4 - x² for x ≤ 2.

a) f^(-1)(x) = √(4 - x) for x ≤ 2
b) f^(-1)(x) = -√(4 - x) for x ≤ 2
c) f^(-1)(x) = 2 - √x for x ≤ 4
d) f^(-1)(x) = 2 + √x for x ≤ 4

Answer 1

The inverse of the function f(x) = 4 - x² for x ≤ 2 is f^(-1)(x) = √(4 - x) for x ≤ 2.

Step-by-step explanation:

To find the inverse of the function f(x) = 4 - x² for x ≤ 2, we need to switch the roles of x and y and solve for y. Here are the steps:

1. Replace f(x) with y: y = 4 - x².
2. Swap x and y: x = 4 - y².
3. Solve for y: y² = 4 - x.
4. Take the square root of both sides, we have two possible solutions:
5. y = √(4 - x) or y = -√(4 - x).

So, the correct answer is f^(-1)(x) = √(4 - x) for x ≤ 2 (option a).