Determine all linear function if f(f(x)) = 4x + 3

The standard expression for a linear function
is:

f(x) = ax + b, where
a>0

We'll impose the constraint from enunciation and
we'll get:

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

We'll substitute f(x) by
ax+b

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

We'll remove the brackets and we'll
have:

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

But f(f(x)) = 4x + 3
(2)

We'll put (1) = (2):

a^2*x
+ ab + b = 4x + 3

For te expressions to be equal the
correspondent coefficients have to be equal:

a^2 =
4

ab + b = 3 => b(a+1) = 3
(3)

Because a>0 => a =
sqrt4

a =
2

We'll substitute a in
(3):

b(a+1) = 3

b(2+1) =
3

3b = 3

We'll divide by
3:

b =
1

The linear function f(x) =
2x + 1.