Math Snap

STEP 1

Assumptions1. The function $f(x)$ is given by $f(x)=\frac{x}{x-1}$
. The function $g(x)$ is given by $g(x)=\frac{13}{x^{}-36}$
3. We are asked to find the composition of the functions $f$ and $g$, denoted as $(f \circ g)(x)$

STEP 2

The composition of two functions, $f$ and $g$, is defined as $(f \circ g)(x) = f(g(x))$. This means that we substitute the function $g(x)$ into the function $f(x)$.

STEP 3

Substitute $g(x)$ into $f(x)$ to get the composition of the functions.
$$(f \circ g)(x) = f(g(x)) = \frac{g(x)}{g(x)-1}$$

STEP 4

Now, substitute the given function $g(x)$ into the expression.
$$(f \circ g)(x) = \frac{\frac{13}{x^{2}-36}}{\frac{13}{x^{2}-36}-1}$$

STEP 5

To simplify this expression, we can multiply the numerator and the denominator by $x^{2}-36$ to get rid of the complex fraction.
$$(f \circ g)(x) = \frac{13}{\frac{13}{x^{2}-36} \cdot (x^{2}-36) - (x^{2}-36)}$$

STEP 6

implify the denominator.
$$(f \circ g)(x) = \frac{13}{13 - x^{2} +36}$$

implify the expression further.
$$(f \circ g)(x) = \frac{13}{49 - x^{2}}$$This is the composition of the functions $f$ and $g$, $(f \circ g)(x)$.