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Exploring Functions in Advanced Mathematics

Composite Functions

Definition

Definition of Composite Functions

If y=f(u)y = f(u), and u=g(x)u = g(x), then y=f[g(x)]y = f[g(x)] is called the composite function formed by ff and gg, denoted as f∘gf \circ g.

SymbolTypePronunciation/ExplanationMeaning in This Document
f(u)f(u)Mathematical symbolf of uOuter function, with u as the independent variable
g(x)g(x)Mathematical symbolg of xInner function, with x as the independent variable
f[g(x)]f[g(x)]Mathematical symbolf of g of xComposite function, composition of f and g
f∘gf \circ gMathematical symbolf circle gComposite function notation, read as “f composed with g”

Properties

Composite functions have the following important properties:

  • Domain: The domain of a composite function is the set of x such that g(x)g(x) is defined and f(g(x))f(g(x)) makes sense
  • Commutative Law: Function composition does not satisfy the commutative law: f∘g≠g∘ff \circ g \neq g \circ f
  • Associative Law: Function composition satisfies the associative law: (f∘g)∘h=f∘(g∘h)(f \circ g) \circ h = f \circ (g \circ h)

Examples

Common examples of composite functions:

  • f(x)=sin⁡(x2)f(x) = \sin(x^2) is the composite of f(u)=sin⁡uf(u) = \sin u and g(x)=x2g(x) = x^2
  • f(x)=exf(x) = e^{\sqrt{x}} is the composite of f(u)=euf(u) = e^u and g(x)=xg(x) = \sqrt{x}
  • f(x)=ln⁡(x2+1)f(x) = \ln(x^2 + 1) is the composite of f(u)=ln⁡uf(u) = \ln u and g(x)=x2+1g(x) = x^2 + 1

Methods for Finding the Domain of Composite Functions

  1. First Step: Find the domain of the inner function g(x)g(x)
  2. Second Step: Find the domain of the outer function f(u)f(u)
  3. Third Step: Find the range of x such that the value of g(x)g(x) belongs to the domain of f(u)f(u)

Derivative of Composite Functions

The derivative of composite functions can be found using the chain rule:

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Exercises

Exercise 1

Find the domain of the composite function f(x)=ln⁡(x2+1)f(x) = \ln(\sqrt{x^2 + 1}).

composite functiondomainlogarithmic functionradical function

Approach: We need to consider the domains of the inner function x2+1\sqrt{x^2 + 1} and the outer function ln⁡u\ln u respectively.

Steps:

  1. Domain of the inner function x2+1\sqrt{x^2 + 1}: x2+1≥0x^2 + 1 \geq 0, which holds for all real numbers x
  2. Domain of the outer function ln⁡u\ln u: u>0u > 0, i.e., x2+1>0\sqrt{x^2 + 1} > 0
  3. Since x2+1≥1>0x^2 + 1 \geq 1 > 0, so x2+1>0\sqrt{x^2 + 1} > 0 holds for all real numbers x

Answer:

The domain is R\mathbb{R} (all real numbers).

Exercise 2

Find the domain of the composite function f(x)=sin⁡(ln⁡x)f(x) = \sin(\ln x).

composite functiondomaintrigonometric functionlogarithmic function

Approach: We need to consider the domains of the inner function ln⁡x\ln x and the outer function sin⁡u\sin u respectively.

Steps:

  1. Domain of the inner function ln⁡x\ln x: x>0x > 0
  2. Domain of the outer function sin⁡u\sin u: u∈Ru \in \mathbb{R}, defined for all real numbers
  3. Therefore the domain of the composite function is the domain of the inner function

Answer:

The domain is (0,+∞)(0, +\infty).

Exercise 3

Find the domain of the composite function f(x)=1x−1f(x) = \sqrt{\frac{1}{x-1}}.

composite functiondomainradical functionrational function

Approach: We need to consider the domains of the inner function 1x−1\frac{1}{x-1} and the outer function u\sqrt{u} respectively.

Steps:

  1. Domain of the inner function 1x−1\frac{1}{x-1}: x≠1x \neq 1
  2. Domain of the outer function u\sqrt{u}: u≥0u \geq 0, i.e., 1x−1≥0\frac{1}{x-1} \geq 0
  3. Solve the inequality 1x−1≥0\frac{1}{x-1} \geq 0:
    • When x−1>0x-1 > 0, 1x−1>0≥0\frac{1}{x-1} > 0 \geq 0, which holds
    • When x−1<0x-1 < 0, 1x−1<0\frac{1}{x-1} < 0, which does not hold
  4. So x−1>0x-1 > 0, i.e., x>1x > 1

Answer:

The domain is (1,+∞)(1, +\infty).

Summary

Symbols Appearing in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
f(x)f(x)Mathematical symbolf of xFunction notation, representing a function with x as the independent variable
f(u)f(u)Mathematical symbolf of uOuter function, with u as the independent variable
g(x)g(x)Mathematical symbolg of xInner function, with x as the independent variable
f[g(x)]f[g(x)]Mathematical symbolf of g of xComposite function, composition of f and g
f∘gf \circ gMathematical symbolf composed with gComposite function notation
f′(x)f'(x)Mathematical symbolf prime of xFirst derivative of the function
f′(g(x))f'(g(x))Mathematical symbolf prime of g of xDerivative of the outer function at the value of the inner function
R\mathbb{R}Mathematical symbolBlackboard bold R (Real numbers)Represents the set of real numbers, the set of all real numbers
(a,+∞)(a, +\infty)Mathematical symbolOpen intervalLeft-open right-infinite interval
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