The domain is every input the function accepts — how far the graph spreads horizontally.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Range
The range is every output produced — how far the graph spreads vertically.
Context restricts them
A function counting people cannot take −3 as an input, whatever the algebra allows.
A worked case
y = x² accepts every real input, but its outputs are never negative. Domain all reals, range y ≥ 0.
Domain is read along the x-axis
Project the graph down onto the horizontal axis: everything it covers is the domain. Project it sideways onto the vertical axis for the range. Reading them off a graph is a matter of projection, not calculation.
Open and closed endpoints
A filled circle means the endpoint is included; an open circle means it is not. Arrows mean the graph continues forever. These marks determine whether the interval notation uses square brackets or round ones.
Interval notation
Square brackets include the endpoint, round brackets exclude it: [2, 5) means from 2 up to but not including 5. Infinity always gets a round bracket, because it is not a value that can be reached.
Domains in several pieces
A graph with a gap has a domain in two parts, written with a union: (−∞, 2) ∪ (2, ∞). Gaps come from division by zero or from restrictions in the situation being modelled, and they must be recorded.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Read a Parabola
Priya describes the domain and range of y = x² + 3.
- Step 1
Any number can be squared, so the domain is every real number.
Step 4: Your Turn
Practice makes it stick.
The Floor
Problem 1 of 2
For y = x², what is the smallest possible output?
The Lifted Floor
Problem 2 of 2
For y = x² + 5, what is the smallest possible output?
Spread and Reach
1 of 8
y = x² − 4. Smallest output?
2 of 8
y = −x². Largest output?
3 of 8
y = 3x + 1. Is the domain every real number? 1 for yes, 0 for no.
4 of 8
A function counts children in a class. Can the input be −2? 1 for yes, 0 for no.
5 of 8
y = x² + 9. Smallest output?
6 of 8
Match each idea to what it describes.
Tap a card on the left to start.
7 of 8
y = −x² + 6. Largest output?
8 of 8
A ticket price function takes whole tickets only. Is the domain every real number? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
y = x² + 2. Smallest output?
What You Learned
- The domain is the set of accepted inputs; the range is the set of produced outputs.
- A square is never negative, which puts a floor under its range.
- Context can restrict both beyond whatever the algebra permits.