# Given f(x)=3x+1 prove that f(f(x)) is increasing.

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To prove that a function is increasing or decreasing, we'll have to do the first derivative test. To do the first derivative test, we'll have to determine the result of the composition of f(x) with f(x).

f(x)*f(x) = f(f(x))

We'll substitute in the expression of f(x), the variable x by the expression of f(x).

f(f(x)) = 3f(x) + 1

f(f(x)) = 3(3x+1) + 1

We'll remove the brackets:

f(f(x)) = 9x + 3 + 1

We'll combine like terms:

f(f(x)) = 9x + 4

Since we know the expression of f(f(x)), we can do the first derivative test.

f'(f(x)) = (9x + 4)'

f'(f(x)) = 9

If the first derivative is positive, then the original function is increasing.

**Since the result of the first derivative test is positive, then f(f(x)) is an increasing function.**

f(x) = 3x+1.

To prove that f(f(x) is increasing.

f(f(x)) = 3f(x)+1

f(f(x)) = 3(3x+1) +1

f(f(x)) = 9x+3+1 = 9x+4.

We find the first derivative. If a function is increasing , then its first derivative must be positive .

{f(f(x))}' = {9x+4}'

{f(f(x))} = 9 > x for all x.

Therefore f(f(x)) is increasing for all x.