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Compose two functions

Composition of functions is understood as applying one function to the output of another: (f∘g)(x)=f(g(x))(f\circ g)(x)=f(g(x)). Learners determine and interpret compositions from equations, tables, graphs, or transformation rules, recognize that order generally matters and that composition is not multiplication, and identify the domain as inputs for which both successive evaluations are defined; treatment is limited to familiar real-valued functions rather than abstract or multivariable compositions.

Detailed Explanation: Compose two functions

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

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

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

Suppose

f(x)=2x+3andg(x)=x2−1.f(x)=2x+3 \qquad\text{and}\qquad g(x)=x^2-1.

Find (f∘g)(x)(f\circ g)(x).

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

g(x)=x2−1 g(x)=x^2-1

  1. Substitute g(x)g(x) into ff:

f(g(x))=2(x2−1)+3 f(g(x))=2(x^2-1)+3

  1. Simplify:

(f∘g)(x)=2x2−2+3=2x2+1. (f\circ g)(x)=2x^2-2+3=2x^2+1.

Therefore,

(f∘g)(x)=2x2+1.\boxed{(f\circ g)(x)=2x^2+1}.

For example, when x=2x=2:

g(2)=22−1=3g(2)=2^2-1=3

and then

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

The composition gives the same result:

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

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

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

which is generally not equal to 2x2+12x^2+1. Since both functions are defined for every real number, the domain of each composition is all real numbers.

Learn by doing: Compose two functions

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    Function Composition - Inputs to Composite Function

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