The problem asks us to find $f(g(h(x)))$ given the following functions:

1. $f(x) = x^4 + 5$
2. $g(x) = x - 5$
3. $h(x) = \sqrt{x}$

Let's go through the steps to find $f(g(h(x)))$:

Step 1: Evaluate $h(x)$

We have: $h(x) = \sqrt{x}$ So, $h(x)$ just takes $x$ and returns $\sqrt{x}$.

Step 2: Substitute $h(x)$ into $g(x)$ to find $g(h(x))$

Since $g(x) = x - 5$, substitute $h(x)$ for $x$ in $g(x)$: $g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 5$

Step 3: Substitute $g(h(x))$ into $f(x)$ to find $f(g(h(x)))$

Now, we take $g(h(x)) = \sqrt{x} - 5$ and substitute it into $f(x) = x^4 + 5$: $f(g(h(x))) = f(\sqrt{x} - 5) = (\sqrt{x} - 5)^4 + 5$

Simplification

Expanding $(\sqrt{x} - 5)^4$ exactly would require binomial expansion, but if a simplified expression is needed, it would be:

$f(g(h(x))) = (\sqrt{x} - 5)^4 + 5$

This is the expression for $f(g(h(x)))$.

Would you like further simplification, or do you have any additional questions?

Related Questions to Explore

1. How would the solution change if $f(x) = x^3 + 2$ instead?
2. What is $f(g(x))$ directly without $h(x)$?
3. How would you expand $(\sqrt{x} - 5)^4$ using binomial expansion?
4. What happens to $f(g(h(x)))$ if $x$ is a negative number?
5. How does the domain of $h(x)$ affect the domain of $f(g(h(x)))$?

Tip

For nested functions, work from the innermost function outward for clarity and accuracy.