FUNCTIONS 11
Composite Functions
A composite function uses the output of one function as the input of another.
The main idea
The composition of \(f\) and \(g\) is
\[\boxed{(f\circ g)(x)=f(g(x))}.\]The inside function acts first.
1. A simple example
Let
\[g(x)=x+2,\qquad f(x)=x^2.\]Then
\[(f\circ g)(x)=f(x+2)=(x+2)^2.\]The composition is therefore a new function.
↓ \(g\)
\(x+2\)
↓ \(f\)
\((x+2)^2\)
2. The order matters
With the same functions,
\[(f\circ g)(x)=(x+2)^2,\]\[(g\circ f)(x)=x^2+2.\]These are different functions. The graph shows that the two parabolas have different vertices and are not interchangeable.
3. Functions as stages
Composition is easiest to understand as a sequence of operations:
\[x\longrightarrow g(x)\longrightarrow f(g(x)).\]The first output becomes the second input.
↓
first function
↓
intermediate value
↓
second function
↓
final output
4. Evaluating a composite function
If
\[g(x)=2x+1,\qquad f(x)=x^2,\]then at \(x=3\),
\[g(3)=7,\qquad f(7)=49.\]Therefore
\[(f\circ g)(3)=49.\]↓ \(g(x)=2x+1\)
\(7\)
↓ \(f(x)=x^2\)
\(49\)
5. Reading nested functions
For
\[f(g(h(x))),\]start with \(h(x)\), then apply \(g\), then apply \(f\).
\[x\to h(x)\to g(h(x))\to f(g(h(x))).\]\(h\)
↓
then \(g\)
↓
then \(f\)
6. Domain of a composite function
Suppose
\[g(x)=x-1,\qquad f(u)=\sqrt{u}.\]Then
\[(f\circ g)(x)=\sqrt{x-1}.\]Because the square-root input must be non-negative,
\[x-1\ge0\quad\Rightarrow\quad\boxed{x\ge1}.\]The graph begins exactly at \((1,0)\).
7. Composition with inverse functions
If \(f^{-1}\) is the inverse of \(f\), then composition returns the original input:
\[f^{-1}(f(x))=x,\qquad f(f^{-1}(x))=x.\]This is one reason composition is essential when studying inverse functions.
↓ \(f\)
\(f(x)\)
↓ \(f^{-1}\)
\(x\)
8. Why composite functions matter for calculus
Many calculus expressions are compositions. For example,
\[\sin(x^2),\qquad e^{3x+1},\qquad \ln(1+x^2).\]The chain rule differentiates an outer function while accounting for the inner function.
\[\frac{d}{dx}f(g(x))=f'(g(x))g'(x).\]×
rate of inner function
↓
chain rule
Composition means feeding one function into another. Order usually matters, the inside function acts first, and domain restrictions of both stages must be respected. Composite functions lead directly to inverse functions and the chain rule.