Skill: Compose two functions

To compose two functions, apply the inside function first, then use its output as the input of the outside function:

$$(f\circ g)(x)=f(g(x)).$$

This is not multiplication. The symbol $\circ$ means “composition.”

Suppose

$$f(x)=2x+3 \qquad\text{and}\qquad g(x)=x^2-1.$$

Find $(f\circ g)(x)$.

1. Start with the inside function, $g(x)$:

$$g(x)=x^2-1$$

2. Substitute $g(x)$ into $f$:

$$f(g(x))=2(x^2-1)+3$$

3. Simplify:

$$(f\circ g)(x)=2x^2-2+3=2x^2+1.$$

Therefore,

$$\boxed{(f\circ g)(x)=2x^2+1}.$$

For example, when $x=2$:

$$g(2)=2^2-1=3$$

and then

$$f(3)=2(3)+3=9.$$

The composition gives the same result:

$$(f\circ g)(2)=2(2)^2+1=9.$$

The order matters. Reversing the functions gives a different result:

$$(g\circ f)(x)=g(f(x))=(2x+3)^2-1,$$

which is generally not equal to $2x^2+1$. Since both functions are defined for every real number, the domain of each composition is all real numbers.