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...
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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.
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