Let's solve each part of the question systematically, starting with the given functions:

$f(x) = 4x + 5 \quad \text{and} \quad g(x) = 8x$

(a) $(f \circ g)(x)$

The composite function $(f \circ g)(x)$ means applying $g(x)$ first, and then applying $f(x)$ to the result of $g(x)$. Mathematically, this is written as:

$(f \circ g)(x) = f(g(x))$

Now, substitute $g(x) = 8x$ into the function $f(x)$:

$f(g(x)) = f(8x) = 4(8x) + 5$ $= 32x + 5$

So, the composite function is:

$(f \circ g)(x) = 32x + 5$

Domain:

Since $g(x) = 8x$ is defined for all real numbers, and $f(x) = 4x + 5$ is also defined for all real numbers, the domain of $(f \circ g)(x)$ is all real numbers, $\mathbb{R}$.

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Would you like to continue with parts (b), (c), and (d), or do you have questions about part (a)?

Additional related questions:

1. How do you find the domain of composite functions in general?
2. How does the composition $g \circ f$ differ from $f \circ g$?
3. What are some examples of real-world situations where composite functions are used?
4. What would happen if the functions were piecewise?
5. How does function composition relate to function inverses?

Tip: Always remember that the domain of a composite function depends on the domain of the inner function and the validity of the outer function when evaluated at points in the inner function's range.