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.
See how the composite function maps directly from domain set \(X\) to codomain set \(Z\) across the intermediate set \(Y\).
A composite function combines two functions into a single operation.
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 \(g(x)\).
In general, \(fg(x) \neq gf(x)\). The order of composition changes the result unless the functions are specially related.
Repeated application of the same function gives \(f^2(x) = f[f(x)]\), \(g^2(x) = g[g(x)]\), and in general \(f^n(x) = f[f^{n-1}(x)]\).
Composite functions link sequential dependencies, such as time determining the radius of a circle, which in turn determines its surface area.
Composite function
\(g\) acts first, then \(f\) acts on \(g(x)\).
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: \(f(x)=2x\), \(g(x)=x^2-5\)
\(gf(x) = 4x^2 - 5\)
Iterated functions
\(g^2(x) = x^4 - 10x^2 + 20\)
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 \(g(x)\).
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 \(f(x)\).
Objective: Identify composite functions, iterated functions, and their properties.
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 \(g(x)\)?
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 _______.
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.
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