find f of g of h, f(X)= absolute value of x-f, g(x)=2^x, g(h) = square root of x

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.

find f of g of h, f(X)= absolute value of x-f, g(x)= 2^x, g(h) = square root of x

Learn how to find f(g(h(x))) when f(x) = |x - f|, g(x) = 2^x, and h(x) = √x. This problem covers key mathematical concepts such as function composition, absolute value, and exponential functions. Suitable for high school students in Grades 9-12.

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