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# How do you find (f o g)(x) and its domain, (g o f)(x) and its domain, (f o g)(-2) and (g o f)(-2) of the following problem ##f(x) = x^2 – 1##, ##g(x) = x + 1##?

Given ##color(white)("XXX")f(color(blue)(x))=color(blue)(x)^2-1## and ##color(white)("XXX")g(color(red)(x))=color(red)(x)+1##

Note that ##(f@g)(x)## can be written ##f(g(x))## and that ##(g@f)(x)## can be written ##g(f(x))##

##(f@g)(x) = f(color(blue)(g(x))) = color(blue)(g(x))^2-1## ##color(white)("XXXXXX")=(color(blue)(x+1))^2-1## ##color(white)("XXXXXX")=x^2+2x## Since this is defined for all Real values of ##x##, the of ##(f@g)(x)## is all Real values. (although it wasn't asked for, the would be ##[-1,+oo)##)

Similarly ##(g@f)(x)=g(color(red)(f(x)))+1## ##color(white)("XXXXXX")=g(color(red)(x^2-1))## ##color(white)("XXXXXX")=color(red)(x^2-1)+1## ##color(white)("XXXXXX")=x^2## Again, this is defined for all Real values of ##x## so the Domain is all Real values. (but the Range is ##[0,+oo)##)