f(x)=3ˣ, g(x)=log₃(x).
(a) f(g(x))=3^(log₃(x))=x
(b) g(f(x))=log₃(3ˣ)=xlog₃(3)=x×1=x
(c) Let y=f(x)=3ˣ, then when we take logs to base 3 of each side:
log₃(y)=x, so x=log₃(y) is the inverse of y=3ˣ.
But g(y)=log₃(y), therefore x=g(y). However, g(x)=log₃(x), because it doesn’t matter what the letter is, it’s only a marker to indicate the argument of the function. Therefore, g(x) is the inverse of f(x), g(x)=f⁻¹(x) and f(x)=g⁻¹(x).
(f o g)(x), the same as f(g(x)), = (gf)(x)=x.