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

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An increasing linear function is of the form y = ax + b where a is positive.

It is given that f(f(x))=4x+3

=> f(ax + b) = 4x + 3

=> a(ax + b) + b = 4x + 3

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

=> a^2 = 4 and ab + b= 3

=> a = 2, a = -2 is rejected as a is positive.

=> 2b + b = 3

=> 3b = 3

=> b = 1

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

f(x) = ax + b; linear function

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