How to do compositions of functions

Question

I've been given a problem that says to find $[f \circ g](x)$ and $[g\circ f](x)$. First off, what is the difference between the two? Second, how might I do this?

Answer 1

The dot symbol means function composition. first apply the right most function, and then apply the left function on the result.

Example: $$f(x)=x^2$$ $$g(x)=2x$$

Then

$$(f\circ g) (x)= f(g(x))=f(2x)=(2x)^2=4x^2$$ $$(g\circ f) (x)= g(f(x))=g(x^2)=2x^2$$

Answer 2

The difference between $(f\circ g)(x)$ and $(g\circ f)(x)$ is simply the order in which the functions are.

For example, if $f(x)=x^2$ and $g(x)=2^x$, then $(f\circ g)(x)=f(g(x))=f(2^x)=(2^x)^2=2^{2x}$

On the contrary, $(g\circ f)(x)=g(f(x))=g(x^2)=2^{x^2}$

More simply, $f\circ g=f(g)$ and $g\circ f=g(f)$