What is the increasing linear function if f(f(x))=4x+3 ?

What is the increasing linear function if
f(f(x))=4x+3

f(f(x)) =
4x+3.

We assume that f(x) is a linear function of the form
ax+b.

f(f(x)) =
a*f(x)+b.

f(f(x)) = a(ax+b)
+b.

=> f((x)) =
a^2x+ab+b...(1)

Also given f(f(x) =
4x+3...(2_

Therefore from (1) and
(2):

 a^2x+ab+b =
4x+3.

Equating like terms: a^2x+4x. So a^2 = 4, a = sqr4 =
2. Or a = -sqrt4 = -2.

Also  ab+b =
3.

When a = 2,  ab+b = 3 gives 2b+b = 3. So 3b=3, or b=
1.

When a= -2, ab+b = 3 gives -2b+b = 3, -b = 3, or b =
-3.

So f(x) ax+b = 2x+1, or f(x) =
-2x-3.

Therefore f(x) 2x+1 is the increasing function as
f'(x) = (2x+1)' = 2 > 0.