# What is the continuity of the composite function f(g(x)) given f(x)=1/(x-6) and g(x)=x^2+5?

The function $f \left(g \left(x\right)\right)$ means to take color(blue)(g(x) and plug it into $f \left(\textcolor{b l u e}{x}\right)$.
So, if $f \left(\textcolor{b l u e}{x}\right) = \frac{1}{\textcolor{b l u e}{x} - 6}$, then $f \left(\textcolor{b l u e}{g \left(x\right)}\right) = \frac{1}{\textcolor{b l u e}{g \left(x\right)} - 6}$. Using color(blue)(g(x)=x^2+5 this becomes $f \left(g \left(x\right)\right) = \frac{1}{\left({x}^{2} + 5\right) - 6} = \frac{1}{{x}^{2} - 1}$.
So, we want to examine the continuity of the function $\frac{1}{{x}^{2} - 1}$.
The only real issue that may arise from this function is if we have a denominator that equals $0$, since that is not possible. Setting the denominator to $0$ to see when this occurs, that yields ${x}^{2} - 1 = 0$, so ${x}^{2} = 1$, so there are discontinuities at $x = 1$ and $x = - 1$.