We are tasked with finding $f(g(x))$, where:

• $f(x) = \frac{x^3 + 10}{x - 4}$
• $g(x) = \sqrt{3x - 5}$

To find $f(g(x))$, we will substitute $g(x)$ into $f(x)$.

Step-by-step solution:

1. Substitute $g(x) = \sqrt{3x - 5}$ into $f(x)$:

   $f(g(x)) = f\left( \sqrt{3x - 5} \right)$

   Since $f(x) = \frac{x^3 + 10}{x - 4}$, we replace $x$ in $f(x)$ with $g(x)$:

   $f(g(x)) = \frac{\left( \sqrt{3x - 5} \right)^3 + 10}{\sqrt{3x - 5} - 4}$

2. Simplify the numerator:

   The cube of $\sqrt{3x - 5}$ is:

   $\left( \sqrt{3x - 5} \right)^3 = (3x - 5)^{3/2}$

   So the numerator becomes:

   $(3x - 5)^{3/2} + 10$

3. Simplify the denominator:

   The denominator is already in a simplified form:

   $\sqrt{3x - 5} - 4$

Thus, the final expression for $f(g(x))$ is:

$f(g(x)) = \frac{(3x - 5)^{3/2} + 10}{\sqrt{3x - 5} - 4}$

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Would you like further explanation or clarifications on this?

Here are some related questions you might find interesting:

1. What is the behavior of $f(x)$ as $x$ approaches 4?
2. How do we find the domain of the composite function $f(g(x))$?
3. What is the value of $f(g(x))$ when $x = 5$?
4. How can we differentiate the composite function $f(g(x))$?
5. What is the inverse of $f(x)$ and how does it relate to $g(x)$?

Tip: To find the domain of a composite function like $f(g(x))$, first ensure the domain of $g(x)$ is valid, and then verify if the output of $g(x)$ is within the domain of $f(x)$.