Domain and range on ACT Math

The domain of a function is the set of inputs you are allowed to use, and the range is the set of outputs the function can actually produce. On the ACT, domain questions almost always come down to two forbidden moves, and range questions almost always come down to reading a graph carefully. Neither one requires clever algebra.

Inputs go in, outputs come out

Think of a function as a machine. The domain is the list of things the machine will accept without breaking. The range is the list of things it can hand back. If f(x) = 2x + 1, you can put in any real number and you can get out any real number, so both the domain and the range are all real numbers. Most functions are that easy, which is exactly why the ACT only bothers to ask when something restricts them.

Since the whole topic is stated in terms of inputs and outputs, it sits right on top of function notation. If f(3) = 7 feels shaky, start there.

Restriction one: you cannot divide by zero

Whenever a variable appears in a denominator, set the denominator equal to zero, solve, and throw those values out of the domain.

• For f(x) = 1/(x - 3), set x - 3 = 0 to get x = 3. The domain is all real numbers except 3.
• For g(x) = (x + 2)/(x^2 - 16), factor the bottom into (x - 4)(x + 4). It equals zero at x = 4 and at x = -4, so the domain is all real numbers except 4 and -4.

The numerator never matters here. Students sometimes exclude x = -2 in that second example because it makes the top zero, but a zero output is perfectly legal. Only a zero denominator breaks the machine.

Restriction two: no square roots of negatives

Inside a square root, the expression has to be greater than or equal to zero. Set up that inequality and solve it.

• For h(x) equal to the square root of (x - 5), you need x - 5 to be greater than or equal to 0, so the domain is all x greater than or equal to 5.
• For k(x) equal to the square root of (8 - 2x), you need 8 - 2x to be greater than or equal to 0. Adding 2x to both sides gives 8 greater than or equal to 2x, so x is less than or equal to 4.

Sometimes the two restrictions show up together. If p(x) equals the square root of (x + 1) divided by (x - 2), the root demands x greater than or equal to -1 and the denominator forbids x = 2. The domain is all x greater than or equal to -1 except 2. Handle each restriction separately, then combine.

Finding the range without a graph

Squares and absolute values cannot be negative, which is what pushes outputs to one side in most range questions.

• For f(x) = x^2 + 3, the smallest x^2 can be is 0, at x = 0. So the smallest output is 3, and the range is all y greater than or equal to 3.
• For g(x) = -(x - 1)^2 + 5, the squared part is again at least 0, but the minus sign flips it, so the largest output happens when the squared part is 0, at x = 1. That gives g(1) = 5, and the range is all y less than or equal to 5.

For any parabola, finding the vertex answers the range question immediately, which is one more reason quadratic equations are worth drilling.

Reading domain and range off a graph

Sweep your eyes horizontally for the domain and vertically for the range. The domain is how far the graph extends left to right along the x-axis. The range is how far it extends bottom to top along the y-axis.

Suppose a curve climbs steadily from the point (-2, 1) to the point (6, 9), with solid dots on both ends. The domain is all x from -2 to 6, and the range is all y from 1 to 9. Watch the endpoints: a filled dot includes that value, an open circle excludes it, and an arrow means the graph keeps going forever that way.

Common traps

• Swapping the axes. This is the number one mistake. Domain lives on the horizontal axis, range on the vertical. If you need a reminder, alphabetical order works: domain before range, x before y.
• Excluding numerator zeros. Only the denominator restricts the domain.
• Forgetting to flip an inequality. If solving your square root condition requires dividing by a negative, the inequality sign reverses.
• Ignoring context in word problems. If x counts students or measures seconds, negative values are out even when the algebra allows them.

A short self-check

Find the domain of f(x) = 5/(2x + 6), the domain of g(x) equal to the square root of (12 - 3x), and the range of h(x) = x^2 - 7. The answers are all real numbers except -3, all x less than or equal to 4, and all y greater than or equal to -7. If you want more reps, the function sets on thirty-six mix these in with the rest of the topic so you practice spotting the restriction rather than being told to look for one. For the bigger picture on how this fits with everything else, see the ACT Math functions hub.

This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.