How Do You Find a Composite Function f(g(x))?

To find the composite function (f(g(x))), substitute the entire expression for $g(x)$ wherever $x$ appears in $f(x)$. Apply $g$ first, then apply $f$ to its output.

The Method

The notation

$$(f \circ g)(x)=f(g(x))$$

means that the output of $g$ becomes the input of $f$. Use these steps:

1. Write down the formula for $f(x)$.
2. Replace every $x$ in that formula with $g(x)$.
3. Use brackets around the substituted expression.
4. Simplify carefully.
5. Check the domain of the resulting composite function.

For example, let

$$f(x)=2x+3 \quad \text{and} \quad g(x)=x^2-1.$$

Substitute $g(x)=x^2-1$ into $f$:

$$f(g(x))=2(x^2-1)+3.$$

Therefore,

$$f(g(x))=2x^2+1.$$

The brackets are essential when the input appears more than once or has a power. For example, if $f(x)=x^2+4x$, then

$$f(g(x))=[g(x)]^2+4g(x).$$

Why Order Matters

A common misconception is that (f(g(x))) and (g(f(x))) are interchangeable. They are generally different because the inner function is applied first.

Using the earlier functions,

$$g(f(x))=(2x+3)^2-1=4x^2+12x+8,$$

which is not equal to $f(g(x))=2x^2+1$.

The domain must also satisfy both stages: $x$ must be in the domain of $g$, and $g(x)$ must lie in the domain of $f$.

For a domain example, let $g(x)=x-4$ and $f(x)=\sqrt{x}$. Then . Although is defined for every real , the input to must be non-negative. Therefore , so the composite has domain . This restriction comes from the outer function after substitution.

Before expanding, read the notation from the inside out and state the intermediate output. This makes sign and bracket errors less likely. If either function contains a denominator, square root, or logarithm, test the original input and intermediate value against all restrictions. Simplifying an algebraic expression does not remove those restrictions; record the valid domain with the final composite.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches, SL 2.5, and it may appear on Paper 1 or Paper 2. For the command term Find, show the substitution before simplifying; writing only the final expression may lose method marks. Always preserve the order and use brackets during substitution.