Question

Let $f\left(x\right)=x^2-2x+5$ and $g\left(x\right)=\frac{4}{x-1}$, how do you find $\left(fog\right)\left(3\right)$?

Answer 1

Start by finding $\left(fog\right)\left(x\right)$
To find this function, substitute $x=\frac{4}{x-1}$ That is $g\left(x\right)$ into $f\left(x\right)$
$\Rightarrow (f\circ g)\left(x\right)={\left(\frac{4}{x-1}\right)}^2-2\left(\frac{4}{x-1}\right)+5$
$=\frac{16}{(x-1{)}^2}-\frac{8}{x-1}+5$
Now substitute $x=3$
$\Rightarrow (f\circ g)\left(3\right)=\frac{16}{(3-1{)}^2}-\frac{8}{3-1}+5$
$=\frac{16}{4}-\frac{8}{2}+5=4-4+5=5$.