IF f(f(x)) =4x + 9, Find f(x)

Question

Please show the solution. Thank you.

Answer 1

f(f(x))=4x+9

Let f(x)=ax+b, then f(f(x))=af(x)+b=a(ax+b)+b=a²x+ab+b.

a²x+ab+b=4x+9, so a²=4; b(a+1)=9.

If a=2, 3b=9, so b=3; if a=-2, b=-9.

Therefore f(x)=2x+3 or -2x-9.

CHECK

f(x)=2x+3:

f(f(x))=2f(x)+3=2(2x+3)+3=4x+6+3=4x+9✔️

f(x)=-2x-9:

f(f(x))=-2f(x)-9=-2(-2x-9)-9=4x+18-9=4x+9✔️

So there are two linear solutions for f(x).