1.2 Composite Functions
A composite function is formed by combining two or more functions where the output or image of the first function becomes the input or object of the second function. If an event or process can be broken down into consecutive steps—such as the radius of an oil slick expanding over time and determining its surface area—composite functions provide a direct mathematical model linking the initial variable directly to the final outcome.
Learning Objectives
- Describe the outcome of the composition of two functions \(f\) and \(g\).
- Determine composite functions such as \(fg(x)\), \(gf(x)\), \(f^2(x)\), and \(g^2(x)\) algebraically and through arrow diagrams.
- Determine the image or the object of a composite function given specific inputs or outputs.
- Determine an unknown constituent function when given the composite function and one of the functions.
- Solve real-world and algebraic problems involving composite functions and iterated functions.
Mapping Arrow Diagram of a Composite Function \(fg\)
See how the composite function maps directly from domain set \(X\) to codomain set \(Z\) across the intermediate set \(Y\).
\(X\)
\(x\)\(Y\)
\(y = g(x)\)\(Z\)
\(z = f(y) = f[g(x)]\)Short Explanation
A composite function combines two functions into a single operation.
Definition and Notation
The notation \(fg(x)\) means \(f[g(x)]\), read as “\(f\) composed with \(g\) of \(x\)”. The inner function \(g\) acts first on \(x\), then the outer function \(f\) acts on the resulting image \(g(x)\).
Order Matters
In general, \(fg(x) \neq gf(x)\). The order of composition changes the result unless the functions are specially related.
Iterated Functions
Repeated application of the same function gives \(f^2(x) = f[f(x)]\), \(g^2(x) = g[g(x)]\), and in general \(f^n(x) = f[f^{n-1}(x)]\).
Real-World Link
Composite functions link sequential dependencies, such as time determining the radius of a circle, which in turn determines its surface area.
Composite function
\(fg(x) = f[g(x)]\)\(g\) acts first, then \(f\) acts on \(g(x)\).
| Expression | Meaning | Order of operation |
|---|---|---|
| \(fg(x)\) | \(f[g(x)]\) | \(g\) first, then \(f\) |
| \(gf(x)\) | \(g[f(x)]\) | \(f\) first, then \(g\) |
| \(f^2(x)\) | \(f[f(x)]\) | \(f\) applied twice |
| \(g^2(x)\) | \(g[g(x)]\) | \(g\) applied twice |
Steps to Determine \(fg(x)\) Algebraically
Identify the inner function \(g(x)\).
Substitute the entire expression of \(g(x)\) into every occurrence of \(x\) in \(f(x)\).
Simplify the resulting expression.
State the final composite function \(fg(x)\).
Example: \(f(x)=2x\), \(g(x)=x^2-5\)
\(fg(x) = 2x^2 - 10\)\(gf(x) = 4x^2 - 5\)
Iterated functions
\(f^2(x) = 4x\)\(g^2(x) = x^4 - 10x^2 + 20\)
Try to Answer First
Answer in your mind, then press “Check Answer”.
Given \(f(x) = 3x + 1\) and \(g(x) = 2x - 5\), determine the expression for \(fg(x)\).
Given \(f(x) = 2x - 1\) and \(g(x) = \dfrac{x}{x + 1}\), evaluate \(fg(2)\).
Given \(f(x) = x + 6\) and \(gf(x) = x^2 + 4\), determine the function \(g(x)\).
Common Mistakes
- Performing operations in the wrong order: confusing \(fg(x)\) with \(gf(x)\) by applying the outer function \(f\) first instead of the inner function \(g\).
- Treating \(fg(x)\) as a product of multiplication, i.e. incorrectly calculating \(f(x) \times g(x)\) instead of substituting \(g(x)\) into \(f(x)\).
- When determining an outer function from \(gf(x)\), simply substituting \(x\) into the expression instead of properly defining a temporary substitute variable such as \(u = f(x)\) and expressing \(x\) in terms of \(u\).
- Omitting the domain restrictions, such as \(x \neq 0\) or \(x \neq 1\), carried over from reciprocal or rational functions when forming composite functions.
Concept Misunderstandings
Composition is commutative, so \(fg(x) = gf(x)\) holds true for all functions.
Composition is generally non-commutative: \(fg \neq gf\). The two are only equal for special cases or specific values of \(x\).
\(f^2(x)\) means \([f(x)]^2\).
In the context of functions, \(f^2(x)\) denotes the iterated composite function \(f[f(x)]\), not the algebraic square of the expression \(f(x)\).
Summary
- Composite function \(fg(x) = f[g(x)]\) means \(g\) acts first on \(x\), and \(f\) acts on \(g(x)\).
- Composite function \(gf(x) = g[f(x)]\) means \(f\) acts first on \(x\), and \(g\) acts on \(f(x)\).
- Iterated functions are formed by repeatedly applying the same function: \(f^2(x) = f[f(x)]\), \(f^3(x) = f[f^2(x)]\), and so on.
- To find an unknown inner function \(g(x)\) from \(f[g(x)]\), write the equation using \(f\) with input \(g(x)\) and solve for \(g(x)\).
- To find an unknown outer function \(g(x)\) from \(g[f(x)]\), let \(u = f(x)\), solve for \(x\) in terms of \(u\), and substitute into \(g[f(x)]\).
Short Activity
Objective: Identify composite functions, iterated functions, and their properties.
A. Objective Quiz
1 Given the functions \(f(x) = 2x\) and \(g(x) = x^2 - 5\), what is the expression for \(gf(x)\)?
2 If \(f(x) = x - 2\) and \(fg(x) = 8x - 7\), which of the following is the function \(g(x)\)?
B. Fill in the Blanks
3 Given \(f(x) = x + 2\) and \(g(x) = x^2\), the composite function \(fg(x)\) expressed in terms of \(x\) is _______.
4 For any function \(f\), the notation \(f^2(x)\) represents the composite function _______.
C. Matching / Drag and Drop
Match each composite function concept or notation to its correct mathematical meaning or result based on \(f(x) = 2x\) and \(g(x) = x + 3\). Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
