Question

The function f(x)=(x-1)^2-4 is not one-to-one. If you restrict the domain for f(x) to x(less than or equal to symbol)1, what is its inverse function and the domain for the inverse?

Answer 1

To find the inverse function, we need to interchange the roles of x and f(x) and solve for x.

Let y = (x-1)^2 - 4.

Switch the roles:
x = (y-1)^2 - 4.

Now solve for y:
y = (x-1)^2 - 4.

This is the inverse function of f(x).

The domain for the inverse function is the range of f(x) with the domain restricted to x ≤ 1.

To find the range of f(x) with x ≤ 1, we need to find the minimum value of f(x) in that domain.

Taking the derivative of f(x):
f'(x) = 2(x-1).

Setting f'(x) = 0:
2(x-1) = 0,
x - 1 = 0,
x = 1.

So the minimum value occurs at x = 1.

Substituting x = 1 into f(x):
f(1) = (1-1)^2 - 4,
f(1) = -4.

Therefore, the range of f(x) with x ≤ 1 is (-∞, -4].

Therefore, the domain for the inverse function is (-∞, -4].