Question

Given the function f(x)=x^2-2, what is f(f(x))

Answer 1

To find f(f(x)) for the function f(x) = x^2 - 2, replace the x in f(x) with the whole function itself. This results in f(f(x)) = x^4 - 4x^2 + 2.

Step-by-step explanation:

The student is asking to find the composition of the function f(x) = x^2 - 2 with itself, that is, to calculate f(f(x)). To find f(f(x)), we need to substitute the entire function f(x) into itself in the place of x. Here's the step-by-step process:

1. First, write down the function f(x): f(x) = x^2 - 2.
2. Next, substitute f(x) into itself: f(f(x)) = (f(x))^2 - 2.
3. Now replace f(x) in the equation with x^2 - 2: f(f(x)) = ((x^2 - 2)^2) - 2.
4. Expand the square: f(f(x)) = (x^4 - 4x^2 + 4) - 2.
5. Combine like terms: f(f(x)) = x^4 - 4x^2 + 2.

Thus, f(f(x)) = x^4 - 4x^2 + 2 is the composition of the function with itself.