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To solve the problem, we need to evaluate the expression:

\[
A = \frac{(f \circ g)(x)}{(f \circ h)(x)}
\]

### Step 1: Determine \( f \circ g(x) \)
\( f \circ g(x) \) means we substitute \( g(x) \) into \( f(x) \).

Given:
\[
f(x) = 4^x \quad \text{and} \quad g(x) = x + 2
\]

Thus:
\[
f \circ g(x) = f(g(x)) = f(x + 2) = 4^{x+2}
\]

### Step 2: Determine \( f \circ h(x) \)
Similarly, \( f \circ h(x) \) means we substitute \( h(x) \) into \( f(x) \).

Given:
\[
h(x) = x - 1
\]

Thus:
\[
f \circ h(x) = f(h(x)) = f(x - 1) = 4^{x-1}
\]

### Step 3: Compute the Expression \( A \)
Now, substitute the expressions for \( f \circ g(x) \) and \( f \circ h(x) \) into \( A \):

\[
A = \frac{4^{x+2}}{4^{x-1}}
\]

### Step 4: Simplify the Expression
To simplify, use the property of exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

\[
A = 4^{(x+2) - (x-1)} = 4^{x+2-x+1} = 4^{3} = 64
\]

Thus, the correct answer is \( \boxed{64} \).

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What would the expression \( \frac{f(x)}{f(h(x))} \) look like for different functions \( f(x) \)?
2. How would the answer change if \( g(x) \) were \( x - 2 \) instead of \( x + 2 \)?
3. What is the significance of the exponent properties in simplifying expressions?
4. How would you evaluate \( A \) if \( f(x) \) were a logarithmic function instead of an exponential function?
5. Can you determine \( A \) if \( h(x) \) was a quadratic function, say \( h(x) = x^2 - 1 \)?

**Tip:** Remember to apply the properties of exponents to simplify expressions quickly, especially when dealing with exponential functions.

Learn how to evaluate the mathematical expression A = ((f ◦ g)(x)) / ((f ◦ h)(x)) using exponential functions and properties of exponents. This problem involves composition of functions and simplifying exponential expressions. Suitable for advanced high school students.

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