Composite Functions

Function Nesting: A composite function, often called a 'function of a function', is formed when one function is substituted into the independent variable of another.

Notation Systems: Composition is commonly written as $fg(x)$, $f(g(x))$, or $(f \circ g)(x)$. All of these notations indicate that $f$ is the 'outer' function and $g$ is the 'inner' function.

Input-Output Chain: The process begins with an input $x$, which is processed by the inner function $g$ to produce $g(x)$. This intermediate result then serves as the input for the outer function $f$, resulting in the final output $f(g(x))$.

The Order Principle: The function closest to the variable $x$ is always applied first. In the expression $fg(x)$, $g$ is the primary operator, and $f$ is the secondary operator.

Non-Commutativity: Function composition is generally not commutative, meaning $fg(x) \neq gf(x)$ for most functions. Changing the order of application usually results in a completely different mathematical expression.

Identity Composition: If $f^{-1}(x)$ is the inverse of $f(x)$, then the composition $ff^{-1}(x)$ or $f^{-1}f(x)$ always simplifies to the identity $x$, effectively 'undoing' the transformation.

The Substitution Method: To find the algebraic expression for $f(g(x))$, replace every instance of $x$ in the formula for $f$ with the entire expression of $g(x)$.

Evaluating Numerical Values: When calculating $f(g(2))$, first evaluate the inner value $k = g(2)$, then substitute that numerical result into $f$ to find $f(k)$.

Self-Composition: The notation $f^2(x)$ refers specifically to $f(f(x))$, which is the function applied to itself, not the square of the function's output $[f(x)]^2$.

Check Domain Compatibility: In exams, always verify that the range of the inner function $g$ falls within the domain of the outer function $f$. If $g(x)$ produces values that $f$ cannot handle, the composite function is undefined for those $x$.

Work Inside-Out: When solving multi-step compositions like $fgh(x)$, resolve the innermost bracket first and work outward to maintain clarity and avoid algebraic errors.

Notation Precision: Be wary of $f^2(x)$ vs $f(x)^2$. Examiners often use these to test if students recognize self-composition versus algebraic squaring.

Verification: To check your algebraic composition, pick a simple number (like $x=1$), evaluate it in the composite expression, and then evaluate it step-by-step using the individual functions to see if they match.