Math Snap

STEP 1

Assumptions1. The function $f(x)$ is given as $f(x)=\frac{5}{ x+1}$
. The function $g(x)$ is given as $g(x)=x^{}-3$
3. We are asked to find the composition of the functions, $f(g(x))$
4. Olivia's work is given as Line1 $f(g(x))=f\left(x^{}-3\right)$ Line $\quad=\frac{5}{ x^{}-3+1}$
Line3 $\quad=\frac{5}{ x^{}-}$

STEP 2

We need to identify where Olivia made a mistake in her work. Let's start by examining Line1.
In Line1, Olivia correctly substituted $g(x)$ into $f(x)$, replacing $x$ in $f(x)$ with $x^{2}-$ from $g(x)$.
$$f(g(x))=f\left(x^{2}-\right)$$This step is correct.

STEP 3

Now let's examine Line2.
In Line2, Olivia substituted $x^{2}-3$ into $f(x)$, replacing $x$ in $f(x)$ with $x^{2}-3$.
$$\frac{5}{2 x^{2}-3+1}$$However, this is not correct. She should have replaced $x$ in $f(x)$ with $x^{2}-3$ in the denominator, not $2x^{2}-3+1$. This is where Olivia made her mistake.

STEP 4

Now let's show the correct way to solve this problem.
First, substitute $g(x)$ into $f(x)$, replacing $x$ in $f(x)$ with $x^{2}-3$ from $g(x)$.
$$f(g(x))=f\left(x^{2}-3\right)$$This is the same as Line1 in Olivia's work.

STEP 5

Next, substitute $x^{2}-3$ into $f(x)$, replacing $x$ in $f(x)$ with $x^{2}-3$.
$$f\left(x^{2}-3\right)=\frac{5}{2\left(x^{2}-3\right)+1}$$

STEP 6

implify the expression in the denominator.
$$f\left(x^{2}-3\right)=\frac{5}{2x^{2}-6+1}$$

Further simplify the expression in the denominator.
$$f\left(x^{2}-3\right)=\frac{5}{2x^{2}-5}$$So, the correct answer is $f(g(x))=\frac{5}{2x^{2}-5}$.