the functions $$ f(x)=\frac{3}{x+4} $$ and $$ g(x)=\frac{13}{x} $$, find the composition $$ f \circ g $$ and simplify your answer as much as possible. Write the domain using int $$ (f \circ g)(x)=\square $$ Domain of $$ f \circ g: \square $$

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Alright, I need to find the composition of the functions $$ f(x) = \frac{3}{x+4} $$ and $$ g(x) = \frac{13}{x} $$, which is $$ f \circ g $$. Let's start by recalling what function composition means. The composition $$ f \circ g $$ is defined as $$ f(g(x)) $$, which means I need to substitute $$ g(x) $$ into $$ f(x) $$ wherever there's an $$ x $$.

So, $$ f(g(x)) = f\left(\frac{13}{x}\right) $$. Now, substituting $$ \frac{13}{x} $$ into $$ f(x) $$, I get:

$$
f\left(\frac{13}{x}\right) = \frac{3}{\frac{13}{x} + 4}
$$

Hmm, that looks a bit complicated. Maybe I should simplify the denominator first. Let's combine the terms in the denominator:

$$
\frac{13}{x} + 4 = \frac{13 + 4x}{x}
$$

So now, the expression becomes:

$$
\frac{3}{\frac{13 + 4x}{x}} = \frac{3x}{13 + 4x}
$$

That seems simpler. But wait, is there a way to factor the denominator to see if it can be simplified further? Let's see:

$$
13 + 4x = 4x + 13
$$

There's no common factor between 13 and 4x, so it seems like the expression $$ \frac{3x}{13 + 4x} $$ is as simplified as it can get.

Now, for the domain of $$ f \circ g $$, I need to consider the domains of both $$ f $$ and $$ g $$. The function $$ g(x) = \frac{13}{x} $$ is undefined when $$ x = 0 $$, so $$ x \neq 0 $$. The function $$ f(x) = \frac{3}{x+4} $$ is undefined when $$ x + 4 = 0 $$, which means $$ x \neq -4 $$.

Since $$ f \circ g = f(g(x)) $$, the domain of $$ f \circ g $$ is the set of all $$ x $$ such that $$ g(x) $$ is in the domain of $$ f $$. So, $$ g(x) $$ must not be equal to -4. Let's set $$ g(x) = -4 $$ and solve for $$ x $$:

$$
\frac{13}{x} = -4 \implies 13 = -4x \implies x = -\frac{13}{4}
$$

So, $$ x = -\frac{13}{4} $$ is also excluded from the domain of $$ f \circ g $$.

Therefore, the domain of $$ f \circ g $$ is all real numbers except $$ x = 0 $$ and $$ x = -\frac{13}{4} $$.

But the question asks to write the domain using intervals. So, the domain can be expressed as:

$$
(-\infty, -\frac{13}{4}) \cup (-\frac{13}{4}, 0) \cup (0, \infty)
$$

Alternatively, in set notation, it's:

$$
\{ x \in \mathbb{R} \mid x \neq 0 \text{ and } x \neq -\frac{13}{4} \}
$$

But since the question specifies to write the domain using intervals, the first expression is more appropriate.

$$ (f \circ g)(x) = \frac{3x}{13 + 4x} $$ Domain of $$ f \circ g: (-\infty, -\frac{13}{4}) \cup (-\frac{13}{4}, 0) \cup (0, \infty) $$

the functions \( f(x)=\frac{3}{x+4} \) and \( g(x)=\frac{13}{x} \), find the composition \( f \circ g \) and simplify your answer as much as possible. Write the domain using int \( (f \circ g)(x)=\square \) Domain of \( f \circ g: \square \)

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