Recovering a missing function

If the outside function is known and one–one, undo that outside rule to recover the inside output. If the inside function is known, rename its output and express the entire composite in that new variable. For f(x)=4x-1 and fg(x)=8x2+3, write 4 g(x)-1=8x2+3. Therefore g(x)=2x2+1. Substitution into f checks it. If g(x)=x+2 and fg(x)=x2+4x+7, put u=x+2. Then x=u-2, so f(u)=(u-2)2+4(u-2)+7=u2+3. Find the range of the known inside function. The composite determines the outside rule only on that range. For an inside rule x2, only non-negative arguments of the outside rule are tested. Check: f(x) is not obtained by dividing the expression for fg(x) by g(x); composition is substitution. If an outside square is undone, its branch must also be justified.