Question

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

Answer 1

The function `f(g(x))` means to take `color(blue)(g(x)` and plug it into `f(color(blue)x)`.

So, if `f(color(blue)x)=1/(color(blue)x-6)`, then `f(color(blue)(g(x)))=1/(color(blue)(g(x))-6)`. Using `color(blue)(g(x)=x^2+5` this becomes `f(g(x))=1/((x^2+5)-6)=1/(x^2-1)`.

So, we want to examine the continuity of the function `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`.