Domain & Range

The domain is the set of inputs a function accepts; the range is the set of outputs it actually produces. In 0606, domain is usually given and range is usually asked, and range questions are sketch questions in disguise.

The method: sketch, then read

For any range question: sketch (or clearly visualise) the graph over the given domain only, then read the output values off the $y$-axis.

$f(x) = x^2 - 6x + 11$ for $x \ge 4$. Find the range. Complete the square: $f(x) = (x - 3)^2 + 2$, vertex at $(3, 2)$. But the domain starts at $x = 4$, to the right of the vertex, where the curve is rising. $f(4) = 1 + 2 = 3$, and $f$ increases from there. Range: $f(x) \ge 3$, not $f(x) \ge 2$.

That last line is the entire trap: the vertex minimum ($2$) is outside the domain, so it’s not in the range. The restricted domain changes the answer, and 0606 sets exactly this configuration repeatedly.

Notation that keeps the marks

• Write ranges in terms of the function: $f(x) \ge 3$ or $3 \le f(x) < 11$, not "$x \ge 3$" and not "$y \ge 3$" ($y$ is tolerated, $x$ is wrong).
• Strict vs non-strict matters: if an endpoint is excluded (open domain end, or an asymptote), the inequality must be strict.
• For exponential and log graphs, the asymptote sets the bound: $f(x) = e^x + 3$ has range $f(x) > 3$, never $\ge$.

Domains you’re expected to spot

When asked for the largest possible domain: exclude division by zero (denominator $\ne 0$), even roots of negatives (expression under $\sqrt{\phantom{x}} \ge 0$), and logs of non-positives (argument $> 0$). State the exclusion as an inequality, that statement is usually the mark.

Common mistakes

• Range read from the formula instead of the domain-restricted sketch
• Vertex value quoted when the vertex lies outside the domain
• Answer in $x$-notation
• Endpoint inclusion/exclusion guessed instead of checked

Domain–range reasoning underpins inverse functions (the inverse swaps them) and the range of a quadratic, secure it here and two other question types come free. Full topic context: Functions notes.