Function Composition Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Given f(x)=x+3f(x) = x + 3 and g(x)=2x2−1g(x) = 2x^2 - 1, find the formula for (g∘f)(x)(g \circ f)(x).

Solution

  1. 1
    (g∘f)(x)=g(f(x))=g(x+3)(g \circ f)(x) = g(f(x)) = g(x + 3).
  2. 2
    Substitute x+3x + 3 into gg: 2(x+3)2−12(x + 3)^2 - 1.
  3. 3
    Expand: 2(x2+6x+9)−1=2x2+12x+18−1=2x2+12x+172(x^2 + 6x + 9) - 1 = 2x^2 + 12x + 18 - 1 = 2x^2 + 12x + 17.

Answer

2x2+12x+172x^2 + 12x + 17
To find a composition formula, replace every xx in the outer function with the entire inner function expression, then simplify.

About Function Composition

Function composition applies one function to the output of another: (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)), meaning evaluate gg first, then apply ff to the result.

Learn more about Function Composition →

More Function Composition Examples

Example 1 easy

Given [formula] and [formula], find [formula].

Example 3 easy

Given [formula] and [formula], find [formula].

Example 4 hard

Given [formula] and [formula], find the domain of [formula].

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Related Concepts

FunctionInverse Function