Proof that f(-x) = -f(x) for f(x) = 3x^3

Introduction
In this problem, we will prove that f(-x) = -f(x) for f(x) = 3x^3.

Substituting -x for x
We start by substituting -x for x in the equation f(x) = 3x^3. This gives us:

f(-x) = 3(-x)^3

Simplifying the expression
To simplify this expression, we first need to evaluate (-x)^3. This can be done by multiplying -x with itself three times:

(-x)^3 = -x * -x * -x = -x^3

Substituting this back into our expression for f(-x), we get:

f(-x) = 3(-x)^3 = 3(-x^3)

Distributing the constant
We can simplify this expression further by distributing the constant 3:

f(-x) = 3(-x^3) = -3x^3

Comparing with f(x)
Now we need to compare this expression with f(x) = 3x^3 to see if f(-x) = -f(x).

f(-x) = -3x^3

We can see that this is indeed the case, since -3x^3 is the same as -f(x).

Conclusion
Therefore, we have proved that f(-x) = -f(x) for f(x) = 3x^3. This means that if we substitute -x for x in the equation 3x^3, we get the opposite of the original value.