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

1 Answer
Nov 15, 2016

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#.