Domain and range from a graph

The domain is every input a function accepts. The range is every output it can produce. Read a graph by looking along the axes: how far left and right does it go, and how far up and down?

y = (x - 1)^{2} - 3

goes on forever left and right, so its domain is all real numbers. Its lowest point is

- 3

, so its range is

y \geq - 3

Formulas that block inputs

Two things break a formula in real numbers: dividing by zero, and the square root of a negative.

Work through it

Find the domain of $f (x) = \frac{5}{x - 2}$ .

The denominator can't be 0. $x - 2 \neq = 0$

So one value is out. $x \neq = 2$

Find the domain of $g (x) = x + 4$ .

What's under the root can't be negative. $x + 4 \geq 0$

Solve. $x \geq - 4$

Watch out for this

Mixing up domain and range.

Domain goes with $x$ , the input. Range goes with $y$ , the output. D comes before R, the way $x$ comes before $y$ .

The short version
Domain: every

x

the graph covers. Range: every

y

it covers. Watch for values that break the formula.

Your turn

Prefer paper? Print the Functions and sequences worksheet, with an answer key.

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