Unit 3 · Topic 2.7
Composition of Functions
Build and interpret composite functions by using one function's output as another function's input. Develop the idea through symbolic, numerical, graphical, and contextual representations.
Learning Goals
- Evaluate compositions from formulas, tables, and graphs.
- Construct and decompose composite functions.
- Determine composite domains.
1. Essential Structure
Build and interpret composite functions by using one function's output as another function's input.
Read the formula together with its domain, units, starting input, and the interval length over which change is measured.
2. Core Ideas
- In \((f\circ g)(x)=f(g(x))\), \(g\) acts first.
- Valid inputs must be in the domain of \(g\), and \(g(x)\) must lie in the domain of \(f\).
- Composition is generally not commutative.
- Input shifts and horizontal scales can be interpreted as composition.
3. Graph and Representation
Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.
4. Original Worked Example
If \(f(x)=x^2\) and \(g(x)=x+1\), find \((f\circ g)(x)\).
Check the result in the original representation and explain its meaning, including units when the quantities are contextual.
5. AP Reasoning Workflow
- Identify the inner function.
- Evaluate or substitute its output.
- Apply the outer function.
- Simplify while preserving restrictions.
- Check the composite domain.
A strong AP response shows the mathematical evidence first and then states a precise conclusion.
6. Extended Example and Application
If \(C(m)=0.85m\) applies a discount and \(T(p)=1.06p\) applies tax, then \((T\circ C)(m)=0.901m\). A fixed fee would make order more visibly important.
7. Technology and Validation
For tables and graphs, read the inner output first and use it as the next input.
When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.
8. Follow the Input-Output Chain
A composition connects functions by feeding the output of one function into the input of another:
The function nearest the input acts first. In \(f\circ g\), this is \(g\), even though \(f\) is written first in the composition notation.
| Expression | First action | Next action |
|---|---|---|
| \(f(g(3))\) | Find \(g(3)\) | Use that result as an input to \(f\) |
| \((g\circ f)(x))\) | Find \(f(x)\) | Substitute the result into \(g\) |
| \(h(g(f(x)))\) | Apply \(f\) | Then \(g\), then \(h\) |
Composition is not multiplication: \(f(g(x))\) is generally different from \(f(x)g(x)\).
9. Compose Functions Given by Formulas
When formulas are available, replace every occurrence of the outer function's input by the entire inner expression. Let
Then
Reversing the order produces
The results demonstrate that composition is usually not commutative. Parentheses protect the inner expression during substitution; dropping them is a common source of sign and exponent errors.
10. Evaluate Composition from a Table
Use the table twice: first read the inner output, then find that number in the input column for the outer function.
| \(x\) | \(f(x)\) | \(g(x)\) |
|---|---|---|
| -2 | 1 | -1 |
| -1 | 3 | 2 |
| 0 | 0 | 3 |
| 1 | 2 | 0 |
| 2 | -1 | -2 |
| 3 | -2 | 1 |
For example, \(g(-1)=2\), so
In the reverse order, \(f(-1)=3\), so \((g\circ f)(-1)=g(3)=1\). If the inner output does not appear among the available inputs for the outer table, the composite value cannot be determined from the table, even if both functions have a listed value at the original input.
11. Evaluate and Construct Graphical Representations
For graphs, trace the same input-output chain without inventing an algebraic formula.
- Locate the starting input \(x=a\) on the graph of the inner function.
- Read or estimate the inner output \(g(a)=b\).
- Move to input \(x=b\) on the graph of the outer function.
- Read the final output \(f(b)=c\), so \((f\circ g)(a)=c\).
To construct points on the composite graph, repeat the process for several starting inputs and plot the pairs \((a,c)\). The intermediate value \(b\) guides the calculation but is not a coordinate of \(f\circ g\).
12. Determine the Domain of a Composition
The domain of \(f\circ g\) contains inputs that pass two tests:
Example 1: If \(f(u)=1/(u-4)\) and \(g(x)=x^2\), then
The inner function accepts every real number, but its output cannot equal 4 because 4 is excluded from the domain of \(f\). Thus \(x\ne-2,2\), and the domain is \(( -\infty,-2)\cup(-2,2)\cup(2,\infty)\).
Example 2: If \(p(u)=\sqrt{u-1}\) and \(q(x)=3/(x-2)\), require both \(x\ne2\) and \(q(x)\ge1\):
A sign analysis gives domain \((2,5]\). Simplifying a formula does not erase restrictions inherited from the component functions.
13. Identity Functions, Decomposition, and Transformations
The identity function \(I(x)=x\) leaves every valid input unchanged:
Decomposition reverses the composition process. For
one useful three-stage decomposition is
There can be more than one valid decomposition. Choose component functions that expose the operations or transformations relevant to the question.
| Transformed function | Composition viewpoint |
|---|---|
| \(f(x+h)\) | \((f\circ s)(x))\), where \(s(x)=x+h\) |
| \(f(kx)\) | \((f\circ d)(x))\), where \(d(x)=kx\) |
| \(f(x)+c\) | \((t\circ f)(x))\), where \(t(u)=u+c\) |
| \(af(x)\) | \((m\circ f)(x))\), where \(m(u)=au\) |
14. Construct a Composite Model in Context
Composition is useful when two quantities are connected through an intermediate variable. Suppose \(F(c)=1.8c+32\) converts Celsius temperature \(c\) to Fahrenheit, and \(C(t)=18-6(0.82)^t\) models a chamber's Celsius temperature \(t\) hours after cooling begins.
The unit chain verifies the order:
At \(t=3\), the model gives approximately \(58.45^{\circ}\text{F}\). The reverse composition \(C(F(t))\) is not meaningful here because \(C\) expects hours, but \(F(t)\) produces a temperature.
AP-ready interpretation: identify the intermediate quantity, justify the composition order using units, write the composite formula, and preserve the contextual domain \(t\ge0\). A formula can be algebraically defined beyond that interval while still being inappropriate for the scenario.
15. Common Errors
- Do not reverse the order: \(f\circ g\) is generally different from \(g\circ f\).
- Multiplying \(f(x)g(x)\) instead of evaluating \(f(g(x))\).
- Ignoring outer-function restrictions on the inner output.
- Giving a numerical result without a domain check, units, or interpretation.
Key Takeaways
- Build and interpret composite functions by using one function's output as another function's input.
- Core relationship: \((f\circ g)(x)=f(g(x))\)
- Error check: Do not reverse the order: \(f\circ g\) is generally different from \(g\circ f\).
- Find \(f\circ g\) and its domain for \(f(x)=\sqrt{x}\), \(g(x)=x-5\).
- Decompose \(h(x)=\ln(3x+1)\).
- Create a context where reversing composition changes the result.