Relate the domain of a function to its graph and to the situation it describes, and choose a function type from the behavior of data

Learning goal: Explain what the domain of a function is, read it from a graph, find it from a formula, and, where the function models a quantitative relationship, decide which inputs the situation actually allows. Then use the behavior of data in a table and the context it comes from to select a type of function, linear, quadratic or exponential, as the model.

Domain. The domain of a function is the set of all inputs for which the function gives an output. When a function is defined only by a formula, the domain is every number the formula can accept. The formula f(x) = 2x + 3 accepts every real number. The formula g(x) = √(x − 1) accepts only x ≥ 1, because a square root of a negative number is not a real number, so the domain of g is x ≥ 1. The formula r(x) = 1/(x − 3) accepts every number except 3, because division by zero is undefined. A domain found this way is called the natural domain of the formula, and it depends on nothing but the arithmetic.

Domain from a graph. On a graph the domain is the set of horizontal positions that the graph occupies. A graph that starts at a filled dot at x = 1 and continues to the right has domain x ≥ 1; a graph drawn only from x = 0 to x = 5 with endpoints included has domain 0 ≤ x ≤ 5; an open circle at an endpoint means that input is excluded. For the rational functions in this unit, a vertical asymptote at x = 3 comes with no point directly above or below x = 3, which is how the graph shows that 3 is missing from the domain. (A function can be given a separately defined value at such an input, so always check whether a point is drawn there.) Reading the domain means asking, for each horizontal position, whether the graph has a point there.

Domain from a situation. When a function models a relationship between two quantities, the situation often allows fewer inputs than the formula does, and the appropriate domain is the set of inputs that make sense for the quantities. Suppose the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, say h(n) = 12n + 8, where the 8 hours are the setup and each engine adds 12 person-hours. The formula 12n + 8 accepts every real number, but a factory assembles a whole number of engines, and assembling zero engines is not an assembly job. The positive integers 1, 2, 3, … are therefore the appropriate domain for h. Its graph is a set of separate points (1, 20), (2, 32), (3, 44), (4, 56), (5, 68) and so on, not a solid line; drawing a line would claim that 2.5 engines is a possible input. This is the difference between a discrete domain, made of separated values such as counts, and a continuous domain, made of every value in an interval, such as time or distance.

Return to the ball whose height is h(t) = −5t² + 20t + 25 meters after t seconds. The formula accepts every real t, and its graph is a full parabola that continues below the horizontal axis on both sides. The situation begins at the throw, t = 0, and ends when the ball lands, at t = 5. The appropriate domain is 0 ≤ t ≤ 5, a continuous interval, because time flows through every value between. Inputs like t = 7 give h(7) = −245 + 140 + 25 = −80, a height 80 meters underground, which the model was never meant to describe. Relating the domain to the situation prevents this kind of nonsense.

A domain restriction changes the graph and the features. Restricting the ball to 0 ≤ t ≤ 5 removes the parts of the parabola with t < 0 and t > 5, which include everything below the axis. What remains is a bounded arc: the rising portion from t = 0 to t = 2 and the falling portion from t = 2 to t = 5. The absolute maximum 45 at t = 2 survives, but there is now also an absolute minimum, 0, at the endpoint t = 5, which the full parabola never had. The end behavior of the full parabola, falling without bound in both directions, disappears entirely, because there are no ends to go to. Domain is not a detail added after the analysis; it decides which features exist.

Choosing a type of function from data and context. Modeling usually starts with data, and the first decision is what kind of function to try. The behavior of the data decides it. Compute the differences between consecutive outputs for equally spaced inputs. If the first differences are constant, a linear model fits the data exactly. If the first differences are not constant but the differences of the differences, the second differences, are constant, a quadratic model fits. If instead consecutive outputs have a constant ratio, an exponential model fits. These tests choose among the three candidate families; a finite table cannot rule out other functions that pass through the same points, so the context has to support the choice. Take three tables for x = 0, 1, 2, 3, 4. The first has outputs 2, 5, 8, 11, 14; the differences are 3, 3, 3, 3, so a linear model y = 3x + 2 fits. The second has outputs 2, 3, 6, 11, 18; the first differences are 1, 3, 5, 7 and the second differences are 2, 2, 2, so a quadratic model fits, here y = x² + 2. The third has outputs 2, 6, 18, 54, 162; the ratios are 6/2 = 3, 18/6 = 3, 54/18 = 3 and 162/54 = 3, so an exponential model y = 2 · 3ˣ fits. Real data are rarely this clean, so the test is whether differences or ratios are approximately constant, and the context must agree: a quantity that grows by a fixed percent each period, such as a population with a steady percent growth rate or an account paying compound interest, calls for an exponential model; a quantity that changes by a fixed amount each period calls for a linear model; a quantity such as area, or the height of a projectile, whose formula involves a square calls for a quadratic model.

The misconception to avoid. Students often assume that the domain of a model is the natural domain of its formula. Both are called domains, but the natural domain describes the formula and the model's domain describes the situation. A cost function C(m) = 2.50 + 1.75m for a taxi ride of m miles has natural domain all real m, but a ride of −4 miles does not exist, so the model's domain is m ≥ 0. If the meter charges by the whole mile, the actual distance is still continuous (a ride can be 2.5 miles long); what changes is the cost rule, which becomes a step function of m, unless the input is redefined as the whole number of billed miles, whose domain is then the nonnegative integers. Choosing the domain is part of choosing the model, and a model without a stated domain is incomplete.

Worked domain from a formula and a graph: Let g(x) = √(x − 1) + 2. The expression under the root must be nonnegative, so x − 1 ≥ 0 and x ≥ 1: the domain is x ≥ 1. Compute g(1) = 2, g(2) = 3, g(5) = 4 and g(10) = 5. The graph starts at the filled endpoint (1, 2) and rises to the right forever, with nothing to the left of x = 1, which is the graph's way of showing the same domain. The range, the set of outputs, is y ≥ 2, because the square root is never negative.

Worked domain from context: A hypothetical school sells tickets to a play at 8 dollars each, and the revenue for n tickets is R(n) = 8n. The formula accepts every real number, but tickets are sold in whole numbers and the theater holds 350 people, so the appropriate domain is the integers from 0 to 350. The graph is 351 separate points from (0, 0) to (350, 2800). The maximum revenue, 2,800 dollars, exists only because the domain has an upper endpoint; the formula 8n alone has no maximum.

Worked choice of model: In a hypothetical experiment, a tank is drained and the depth of water, in centimeters, is measured each minute: 100, 81, 64, 49, 36, 25 at t = 0 through 5. The first differences are −19, −17, −15, −13, −11, not constant, so the model is not linear. The second differences are 2, 2, 2, 2, constant, so a quadratic model is appropriate; the ratios 81/100 = 0.81 and 64/81 ≈ 0.79 are not constant, so an exponential model is not. The values are (10 − t)², so d(t) = (10 − t)² fits every entry: d(3) = 49 and d(5) = 25 check. The context sets the domain as 0 ≤ t ≤ 10, because at t = 10 the tank is empty and the formula would start rising again, which no draining tank does.

Representation task: For h(n) = 12n + 8, the person-hours to assemble n engines, draw two graphs side by side: one showing the formula on every real input as a solid line, and one showing only the points for the positive integers 1 through 8. Under the second graph write why the positive integers are the appropriate domain, using the words discrete, count and situation. Then do the same for the ball height h(t) = −5t² + 20t + 25 with the domain 0 ≤ t ≤ 5, and explain why this domain is continuous rather than discrete.

Independent performance: Write three functions of your own, each modeling a real quantity: one whose appropriate domain is a set of nonnegative integers, one whose domain is a bounded interval of real numbers, and one whose formula excludes some input for arithmetic reasons (a root or a division). For each, state the natural domain of the formula, the domain the situation allows, and one input that the formula accepts but the situation rejects, with the nonsense output it would give. Then collect or invent a table of six equally spaced data values, compute first differences, second differences and ratios, and select a linear, quadratic or exponential model, justifying the choice from the numbers and the context. Success criteria: each domain is stated with correct inequality symbols or the word integers; the rejected input is explained in terms of the quantities; the differences and ratios are computed correctly; and the chosen model type is the one the data support.