Properties of the Range

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Finding a range means establishing both a restriction and attainability. Showing that outputs cannot be negative is only half the argument; you must also show which nonnegative outputs occur.

Method

Useful methods include completing the square, monotonicity on the stated domain, and solving $y=f(x)$ for an allowed $x$ . Endpoint inclusion follows from whether an input actually attains the candidate output.

Worked example

For $f(x)=(x-2)^2+3$ on $\mathbb R$ , the square gives $f(x)\ge3$ . Conversely, for any $y\ge3$ , choose $x=2+\sqrt{y-3}$ , which produces $f(x)=y$ . The range is exactly $[3,\infty)$ . For $1/x$ , output zero is impossible, while every nonzero $y$ is attained by $x=1/y$ .

Common mistake

A graph window can suggest a range but cannot prove behavior outside its visible portion.

Check your understanding

Find the range of $f(x)=5-2x^2$ on $\mathbb R$ .

Show answer

$(-\infty,5]$ , with maximum $5$ attained at $x=0$ .

Explore

Move and scale a graph

Try this. Change one slider at a time. Positive h moves the vertex right; k moves it up; negative a reflects the curve and a = 0 makes it constant.

Vertical multiplier a1Horizontal shift h0Vertical shift k0

y = 1(x − (0))² + (0). Vertex (0, 0); opens up. 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.

If f: A → B is a function, which statement must hold?

Its range equals A.

Its range is a subset of B.

Its range equals B.

Hint 1 · Find a starting point

The codomain includes all allowed outputs.

Hint 2 · Take the next step

Some allowed outputs may never be produced.

Show the reasoning

Answer: Its range is a subset of B.

Every actual output lies in B, but equality requires f to be onto.

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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