Domain of a Composite Function

The domain of a composite is the set of inputs for which the whole expression is defined. When the composite contains a square root, the quantity under the root must be $\ge 0$; that condition gives the domain.

The functions $f$ and $g$ are defined by $f(x) = \sqrt{x}$ for $x \ge 0$, and $g(x) = x - 3$. Find $fg(x)$ and state its domain. [4]

The working

Step 1, form $fg(x) = f\big(g(x)\big)$ (apply $g$ first, then $f$): $fg(x) = f(x - 3) = \sqrt{x - 3} \quad \text{(M1, A1)}$

Step 2, find the domain. The expression under the square root must be $\ge 0$: $x - 3 \ge 0 \;\Rightarrow\; x \ge 3 \quad \text{(M1, A1)}$

So $fg(x) = \sqrt{x - 3}$, defined for $x \ge 3$.

Where the marks are won and lost

• Composite order: $fg$ means $g$ first, so $f$ acts on $(x - 3)$, giving $\sqrt{x - 3}$, not $\sqrt{x} - 3$.
• The square-root condition ($x - 3 \ge 0$) is what fixes the domain. Forgetting it, or writing $x \ge 0$ (the domain of $f$ alone), loses the domain marks.
• The domain of the composite comes from the composite, not just from $f$‘s or $g$‘s individual domains.

Common mistakes

• Writing $fg(x) = \sqrt{x} - 3$ (wrong order, $f$ applied first).
• Giving the domain as $x \ge 0$ instead of $x \ge 3$.
• Using strict $>$ when $\ge$ is correct ($\sqrt{0}$ is defined).

Full method: Composite Functions notes. See also Domain & Range. Topic home: Functions pillar.