What is the difference between domain and range?

The domain is the set of inputs a function accepts. The range is the set of outputs it produces from those inputs.

Should I use f(x) or y when writing the range?

Write it in terms of f(x), for example f(x) ≥ 2. Writing x ≥ 2 describes inputs, not outputs, and loses the mark.

How to Find the Range of a Function Algebraically

Short answer

How do you find the range of a function?

The range is every output the function can give. For a quadratic, complete the square to find its minimum or maximum. For a restricted domain, also check the values at the ends. For a fraction, find the value the output can never reach.

The range of a function is the set of all its outputs. You find it by asking: what is the smallest output, the largest output, and is any value impossible?

A quadratic: complete the square

Find the range of $f (x) = x^{2} - 6 x + 11$ for $x \in R$ .

f (x) = (x - 3)^{2} + 2

A square is never negative, so the smallest value of $(x - 3)^{2}$ is 0, at $x = 3$ . The smallest output is therefore 2.

Range: f (x) \geq 2

A restricted domain: check the ends too

Same function, but for $0 \leq x \leq 5 $. T h e minim u mi ss t i l l in s i d e t h e d o main, a t$ x = 3 $, w h er e$ f(3) = 2 $. N o w c h ec k b o t h e n d s :$ f(0) = 11 $an d$ f(5) = 6$. The largest of these is 11.

Range: 2 \leq f (x) \leq 11

For a straight line the ends are all you need. For $f (x) = 5 - 2 x$ with $1 \leq x \leq 4 $:$ f(1) = 3 $an d$ f(4) = -3 $, so$ -3 \leq f(x) \leq 3$.

A fraction: find the impossible output

For $g (x) = \frac{1}{x - 2}$ , a fraction with numerator 1 can never equal 0. Every other value is possible.

Range: g (x) \neq = 0

A square root: the smallest value is at the start

For $h (x) = x - 2 + 1$ , the square root is at least 0, so $h (x) \geq 1$ .

Try it: find which inputs the function accepts

Domain and range go together: the range comes from feeding every allowed input through.

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Mistakes that cost marks

Writing the range in terms of $x$ . Use $f (x)$ .
Only checking the ends of a restricted quadratic, and missing a turning point inside.
Using $>$ when the value is reached. $f (3) = 2$ is reached, so it is $\geq 2$ .

Next: how to find the inverse of a function, whose domain is this range.

How to find the range of a function