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

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You need to evaluate the function f(f(x)) such that:

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

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

The function f(f(x)) = 9x + 4 is a linear function, whose leading coefficient is positive, hence f(f(x)) = 9x + 4 increases over R.

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.