Domain and Range

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A function is only as good as the inputs it accepts. The domain is the set of inputs $x$ for which $f(x)$ is defined; the range is the set of values $f$ actually produces.

The three classic domain killers

Division by zero: $f(x)=\frac{1}{x-2}$ excludes $x=2$ .
Even roots of negatives: $f(x)=\sqrt{x+3}$ requires $x+3 \ge 0$ .
Logarithms of non-positives: $f(x)=\ln(5-x)$ requires $x < 5$ .

Worked pattern

Find the domain of $f(x)=\dfrac{\sqrt{x+3}}{x-2}$ . Require $x+3 \ge 0$ AND $x-2 \ne 0$ . So the domain is $[-3, 2) \cup (2, \infty)$ .

Explore

Use the grapher to see how the graph "ends" at a domain boundary and "breaks" at an excluded point.

Domain and range

Try this. Switch functions and compare allowed inputs and outputs. Open circles exclude endpoints; filled circles include them.

Function

y = x² · Domain: all real x · Range: y ≥ 0

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What is the real domain of f(x)=√(x−2)?

[2,∞)

(2,∞)

ℝ

Hint 1 · Find a starting point

A real square root needs a nonnegative radicand.

Hint 2 · Take the next step

Solve x−2≥0, retaining equality.

Show the reasoning

Answer: [2,∞)

Zero is an allowed radicand, so x=2 is included.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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