Part 3 of 10 · FunctionsSee the whole course
Domain and range, explained simply
The domain is every input a function can take (the x-values). The range is every output it can give back (the y-values). Picture the graph casting shadows: the shadow on the floor, along the x-axis, is the domain. The shadow on the wall, along the y-axis, is the range.
Two shadows of the same graph
Imagine a lamp far above the graph, shining straight down. The graph throws a shadow onto the x-axis, the floor. Every x that gets a shadow is in the domain.
Now imagine a lamp far to the right, shining straight across. The graph throws a shadow onto the y-axis, the wall. Every y that gets a shadow is in the range.
The picture below does both. Pick a function, then slide the input and watch the dot. When the input is not allowed, the dot disappears.
Range: y ≥ 0
Find them from a list of points
When a function is a list of pairs, the domain is the first numbers and the range is the second numbers. List each number once.
Points: (1, 2), (2, 5), (3, 5)
Domain: {1, 2, 3}
Range: {2, 5}
The output 5 shows up twice, but you only write it once.
Find them from a graph
For the domain, look left to right: which x-values does the graph cover? For the range, look bottom to top: which heights does it reach? Arrows mean the graph keeps going.
Take y = x². The curve goes on forever to the left and to the right, so the domain is all numbers. It never goes below 0 but climbs forever, so the range is y ≥ 0 (y is 0 or more).
Find them from a rule: what can go wrong?
With a rule, assume every number is allowed, then look for the two things that break a rule:
- Dividing by zero. Any input that makes a bottom equal to 0 is out.
- The square root of a negative number. Inputs that put a negative number under a root are out.
y = √x → domain: x ≥ 0, range: y ≥ 0
y = 1/x → domain: x ≠ 0, range: y ≠ 0
A square root is never negative, so the range of √x starts at 0. And 1/x can get as close to 0 as you like but never equals 0, so 0 is missing from its range too. The symbol ≠ means “not equal to”.
Domain and range in real life
A plant is 1 cm tall when it sprouts and grows 2 cm a day for 3 days. Its height is h = 1 + 2d, with d from 0 to 3.
d = 0 → h = 1
d = 3 → h = 7
The domain is the days that make sense: 0 to 3. The range is the heights it reaches: 1 to 7. In real situations, the domain is limited by what is possible, not by the rule.
Your teacher may ask for interval notation. “x ≥ 0” is written [0, ∞), which means “from 0 upward, including 0”. It is the same idea in different clothes.
Watch it explained in under a minute
The five mistakes to avoid
- Mixing the two up. Domain goes with x (what goes in), range goes with y (what comes out). D comes before R, and x comes before y.
- Writing a repeated output twice in the range. List each value once.
- Forgetting the missing input, like 0 for 1/x.
- Reading the domain from the edges of the picture. If the graph has arrows, it keeps going.
- Thinking the range is “how far the graph spreads”. It is the set of outputs the function actually reaches.
Try it yourself
Tap a question to see the answer.
1. Domain and range of the points (0, 3), (1, 3), (2, 7)
Domain: {0, 1, 2}. Range: {3, 7}, because 3 appears twice but is listed once.
2. Which input is not allowed in f(x) = 1 ÷ (x − 2)?
The bottom is 0 when x − 2 = 0, so when x = 2. The domain is all numbers except 2.
3. What is the range of y = x² + 1?
x² is never below 0, so x² + 1 is never below 1. The range is y ≥ 1. The domain is all numbers.
Where domain and range fit in
Domain and range tell you what a function accepts and what it can say back. If you are new to functions, start with what a function is and what f(x) means. Next in this course we move on to lines, starting with slope.