The image shows a problem involving composite functions. Here is the text extracted from the image:

Given: $f(x) = x^2 - 1, \quad g(x) = x - 1, \quad h(x) = \sqrt{x}$

Find: $f(g(h(x)))$

Solution:

To find $f(g(h(x)))$, we will evaluate the functions step-by-step from the inside out.

1. Find $h(x)$: $h(x) = \sqrt{x}$

2. Evaluate $g(h(x))$: Substitute $h(x)$ into $g(x)$: $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 1$

3. Evaluate $f(g(h(x)))$: Substitute $g(h(x))$ into $f(x)$: $f(g(h(x))) = f(\sqrt{x} - 1)$ Now, substitute $\sqrt{x} - 1$ into $f(x)$: $f(\sqrt{x} - 1) = (\sqrt{x} - 1)^2 - 1$

4. Simplify the expression: $(\sqrt{x} - 1)^2 - 1 = (\sqrt{x})^2 - 2\sqrt{x} + 1 - 1 = x - 2\sqrt{x}$

Final Answer:

$f(g(h(x))) = x - 2\sqrt{x}$

Would you like more details on this solution, or do you have any questions?

Related Questions:

1. What is the domain of the function $f(g(h(x)))$?
2. How would you find $g(f(x))$ given the same functions?
3. What would $h(f(g(x)))$ look like?
4. How do you find the inverse of a composite function like $f(g(x))$?
5. How does the composition of functions affect their domains?

Tip:

When dealing with composite functions, always work from the inside out to ensure you substitute and simplify correctly.