i dont understand functions

Question

f(x)=x³, g(x)=x²+3x-9

(f of g)(x)

(g of f)(x)

Answer 1

A function describes the relationship between 2 or more variables. For the simplest case with only 2 variables, one is the independent variable, and the other is the dependent variable. This is a fancy way of saying that one depends on the other.

Suppose we say that f(x) = 2x, What we are really saying is that "f" acts by taking the expression in parentheses and multiplying by 2.

Having defined f, we can write f(y) = 2y, or f(z) = 2z.

Next define another function: g(x) = x + 2

Now, suppose we are given y = f(g(x))

First, plug in our definition for g(x) to get y = f(x + 2)

Then, apply f by multiplying by 2:

y = 2(x + 2) = 2x + 4

What if we are given y = g(f(x)?

f(x) = 2x

g(2x) = 2x + 2

So....f of g of x does NOT equal g of f of x

f(g(x)) ≠ g(f(x))

Answer 2

f(x) = x³ and g(x) = x² + 3x - 9

f of g(x) = f(g(x)) = f(x² + 3x - 9) = (x² + 3x - 9)³

g of f(x) = g(f(x)) = g(x³) = (x³)² + 3(x³) - 9 = x⁶ + 3x³ - 9