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QUESTION

# 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))