To find $f(g(h(x)))$, where:

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

Step 1: Substitute $h(x)$ into $g(x)$

Since $h(x) = \sqrt{x}$, we substitute $h(x)$ into $g(x)$:

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

Step 2: Substitute $g(h(x))$ into $f(x)$

Now we substitute $g(h(x)) = 2^{\sqrt{x}}$ into $f(x)$:

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

Final Answer:

The function $f(g(h(x)))$ is:

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

Would you like any further details or explanation?

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Here are five related questions to explore:

1. How would the function change if $h(x) = x^2$ instead of $h(x) = \sqrt{x}$?
2. What happens to $f(g(h(x)))$ if $x = 0$?
3. How would you evaluate $f(g(h(x)))$ for specific values of $x$?
4. How does the graph of $f(g(h(x)))$ look for different ranges of $x$?
5. Can we solve for $f(g(h(x))) = 0$ for a particular $x$?

Tip: Pay attention to the order of operations when composing functions, as it can greatly affect the final result.