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

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Since given that f(x) is a linear function, then we will assume that:

f(x) = ax+ b

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

==> f(f(x))= (a^2) x + (ab + b) = 4x+3.

Now we will compare both sides.

==> a^2 = 4 ==> a= -+2

But the function is increasing, then, the factor of x should be positive.

Then, we will not consider a= -2.

**==> a= 2**

==> Also, from the equation we know that:

ab+b = 3

==> 2*b + b = 3

==> 3b = 3

**==> b= 1**

**Then, the increasing function f(x) = 2x + 1**

The expression of the linear function is:

f(x) = ax + b

If f(x) is increasing, then the coefficient of x is positive:

a>0

We'll write the expression of f(f(x)), substituting x by f(x) in the expression of f(x):

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:

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

We'll compare and we'll get:

a^2 = 4

a1 = 2 and a2 = -2

Since a has to be positive, we'll reject a2 = -2.

So, a = 2.

ab + b = 3

We'll factorize by b:

b(a + 1) = 3

b(2 + 1) = 3

3b = 3

b = 1

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

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.