Domain and range: how to find them, with a worksheet

What domain and range mean

A function takes an input and gives exactly one output. The domain is the set of all the inputs the function can take, and the range is the set of all the outputs it actually gives. With x and y, the domain is about x and the range is about y.

Think of a vending machine. The buttons you can press are the domain, and the snacks that can come out are the range. Each button gives one snack, which is what makes it a function. Two buttons can give the same snack, but one button can never give two different snacks.

Is it a function?

A relation is any set of input-output pairs. It is a function only if every input has exactly one output, so look for repeated x-values. {(1, 4), (2, 6), (1, 7)} is not a function, because the input 1 has two outputs, 4 and 7. {(1, 4), (2, 4), (3, 4)} is a function, because a repeated output is fine.

On a graph, use the vertical line test. If any vertical line crosses the graph more than once, one x-value has two y-values, and it is not a function. A straight line that is not vertical passes the test; a circle fails it.

Domain and range from ordered pairs and tables

For a list of ordered pairs or a table, the work is careful listing. Write each value once, from smallest to largest, inside curly brackets.

• {(3, 9), (−1, 1), (0, 0), (2, 4)}: domain {−1, 0, 2, 3}, range {0, 1, 4, 9}.
• A table with x = 1, 2, 3, 4 and y = 10, 20, 20, 30: domain {1, 2, 3, 4}, range {10, 20, 30}.
• {(5, −2), (6, −2), (7, −2)}: domain {5, 6, 7}, range {−2}. This is a function with only one output.

Domain and range from a graph

On a graph, the domain is how far the graph reaches from left to right, and the range is how far it reaches from bottom to top. A filled dot at an end means that value is included, an open dot means it is not, and an arrow means the graph keeps going.

• A line segment from (−3, 1) to (5, 9) with filled dots: domain −3 ≤ x ≤ 5, range 1 ≤ y ≤ 9.
• The line y = 2x + 1 with arrows at both ends: the domain and the range are both all real numbers.
• The parabola y = x² − 4: the domain is all real numbers. Its lowest point is (0, −4), so the range is y ≥ −4.

Finding the domain from a formula

When a function is given as a formula, the domain is every real number except the ones that break it. At this level, two things break a function: dividing by zero, and taking the square root of a negative number.

• f(x) = 3x − 7: nothing breaks it, so the domain is all real numbers.
• f(x) = 1/(x − 2): x − 2 cannot be 0, so the domain is x ≠ 2.
• f(x) = √(x + 5): x + 5 must be 0 or more, so the domain is x ≥ −5.
• f(x) = √(6 − x): 6 − x ≥ 0, so the domain is x ≤ 6.
• f(x) = 1/√(x − 1): the root is on the bottom, so it must be more than 0, and the domain is x > 1.

Real situations limit the domain too. If C(n) = 4n is the cost in dollars of n concert tickets, n must be a whole number, 0 or more, because you can't buy half a ticket or −3 tickets.

How it is taught in school

In the US, Common Core grade 8 defines a function as a rule that assigns each input exactly one output (8.F.A.1). High school adds the words domain and range: a function from one set, the domain, to another, the range, assigns each element of the domain exactly one element of the range (HSF-IF.A.1). Students also relate the domain to a function's graph and to the situation it describes (HSF-IF.B.5). Most students meet domain and range in Algebra 1 and return to it in Algebra 2.

In England, GCSE Higher pupils treat expressions as functions with inputs and outputs, and work with inverse and composite functions. The words domain and range, and restricting the domain of a function, are taught formally in A level Mathematics.

How parents can help

• Use the input-output idea in everyday life. Each person has one birthday, so person to birthday is a function, even though many people share a birthday.
• Ask which is the input and which is the output before your child lists anything. Most mix-ups come from swapping x and y.
• Encourage neat lists: curly brackets, each value once, smallest first.
• For formulas, ask the same two questions every time. Could the bottom be zero? Could the inside of a square root be negative?
• Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.