Algebra 1 · F.IF.A.1 · F.IF.B.5

Domain and range from a graph

A graph shows a function as a picture. Which inputs does it use, and which outputs does it ever reach?

You need

Graphs of functionsGrades 8 to 9 Interval and set-builder notationAlgebra 1

Unlocks

A reasonable domain for a real situationAlgebra 1 Graphing by transforming a parent graphAlgebra 2 Graphs of exponential functionsAlgebra 1 Increasing, decreasing, constant, positive and negative intervalsAlgebra 1 The implied domain of a formulaAlgebra 2 +2 more

Where you are 4 in Domain and range

Functions13

What a function is5

Domain and range4

Domain and range from pairs and tables

Domain and range from a graph

A reasonable domain for a real situation The implied domain of a formula

When you reach for it

The domain of a function is the set of inputs it accepts, and the range is the set of outputs it actually produces. For a short list of pairs you collect them, as in domain and range from pairs and tables. A graph usually has infinitely many points, too many to list. What you can read instead is how far the graph stretches: left and right for the domain, down and up for the range.

Reach for this whenever a function is given as a graph. It is the first look before reading a value from a graph: which inputs does the graph cover at all, and which heights does it ever reach? Answers are written as inequalities and in interval notation, which you already have.

The lesson runs over two days. Day 1 reads a graph in one unbroken piece: its reach each way, its end dots and its arrows. Day 2 handles a graph in two pieces, a single missing point, a highest or lowest point in the middle, and a graph that levels off toward a height it never reaches.

The procedure

Day 1: the shadow on each axis

Every point (x, y) on a graph says one thing: the input x gives the output y. So a number a is in the domain exactly when the graph has a point directly above or below a on the x-axis, and a number b is in the range exactly when the graph has a point level with b on the y-axis.

Picture a light shining straight down onto the graph: its shadow on the x-axis is the domain. A light from the side throws a shadow on the y-axis, and that is the range.

Figure 1The domain is the graph’s shadow on the x-axis, the range its shadow on the y-axis.

Figure 1 shows a graph made of straight pieces. It starts at a closed dot at (-4, 1), falls to (-2, -1), climbs to (1, 4), and falls to an open dot at (3, 2).

The domain. The leftmost point is (-4, 1), and the closed dot says it is on the graph, so -4 is in the domain. The rightmost would be (3, 2), but the dot is open, so 3 is not. The graph is unbroken in between, so it passes over every x from -4 to 3:

-4 \le x < 3, \quad\text{that is}\quad [-4, 3).

The range. The lowest point is (-2, -1) and the highest is (1, 4), both on the graph. On its unbroken climb from height -1 to height 4 the graph passes every height between:

-1 \le y \le 4, \quad\text{that is}\quad [-1, 4].

The ends of the domain are always the leftmost and rightmost points. The ends of the range are wherever the graph is lowest and highest, which here is not at its ends at all.

Day 1: closed dots, open dots and arrows

A closed dot means the point is on the graph, so its coordinates are reached: a bracket.
An open dot means the point is not on the graph. The number it would have given is left out, with a parenthesis, unless another point of the graph supplies it. In Figure 1 the open dot at (3, 2) removes x = 3 from the domain, since nothing else sits above 3. It does not remove y = 2 from the range: on the climb from (-2, -1) to (1, 4) the graph passes height 2 (at x = -0.2).
An arrow means the graph carries on forever that way. Carrying on to the right, the domain has no right end and is written with \infty; to the left, with -\infty. If it also carries on up or down, the range has no top or no bottom. An end at \infty or -\infty always takes a parenthesis, since no number is \infty.

The domain is written from left to right and the range from bottom to top, smaller number first, whatever order the graph reaches them in. The domain’s inequality is about the input letter (usually x, or t for time), the range’s about the output letter (usually y).

Day 2: gaps, missing points and levels never reached

A graph in pieces. Each piece throws its own shadow. The domain is all the shadows on the x-axis together, joined with \cup (“or”) where there is a gap; the range is all the shadows on the y-axis together. Shadows that overlap merge, so a gap in the domain need not leave a gap in the range.

A single missing point. An open dot in the middle of a graph removes its input from the domain, unless a closed dot elsewhere in the same column says the output there is somewhere else. It removes its output from the range only if no other point of the graph has that height.

A turning point. The top or bottom of the range is often where the graph turns round, such as the vertex of a parabola. That point is on the graph, so the range includes it. Turning points are the subject of maximums and minimums.

A graph that levels off. Some graphs flatten out toward a height, getting closer and closer without ever reaching it. No dot is drawn, yet that height is not in the range, so the range has an open end there. Such a level is a horizontal asymptote.

Domain and range are separate questions. The domain can be one stretch while the range is in pieces, or endless while the range is short. Each is read from its own shadow.

Worked examples

Example 1

Day 1: a segment with one open end

Find the domain and range of the graph in Figure 2, the segment from the closed dot (-2, 5) to the open dot (3, -5). Write each as an inequality and in interval notation.

Figure 2Example 1. The segment from (-2, 5), included, to (3, -5), not included.

Solution

Domain. The leftmost point is (-2, 5), a closed dot, so x = -2 is in the domain. The rightmost would be (3, -5), but the dot is open, so x = 3 is not. The segment is unbroken between them:

-2 \le x < 3, \quad\text{that is}\quad [-2, 3).

Range. The segment falls, so its highest point is the left end, (-2, 5), which is on the graph. Its lowest would be the right end, at height -5, but that point is missing and, since the segment falls steadily, no other point has height -5. The segment reaches -4.9, -4.99 and every height above -5, but never -5 itself:

-5 < y \le 5, \quad\text{that is}\quad (-5, 5].

The range opens with the parenthesis from the open dot, even though that dot is the graph’s right end. Each bracket comes from the dot at its own end; the order on the page comes from the numbers.

Example 2

Day 1: arrows on a ray and on two lines

Find the domain and range of each graph in Figure 3. In (a) the ray starts at the closed dot (-1, 3) and falls to the right, one unit down for each unit across, without end. In (b) and (c) the lines carry on forever in both directions.

Figure 3Example 2. An arrow means the graph carries on forever in that direction.

Solution

(a) Domain. The graph starts at x = -1, included, and the arrow carries it to the right forever: x \ge -1, that is [-1, \infty).

Range. The highest point is the dot, at height 3. Moving right, the ray falls without end, so it passes every height below 3: y \le 3, that is (-\infty, 3]. One arrow, pointing right and down, opens both the right end of the domain and the bottom of the range.

(b) The line rises without end both ways. Going left it passes every x and drops below every height; going right it passes every x and climbs above every height. The domain is (-\infty, \infty) and so is the range.

(c) The line is horizontal at height -2. Its arrows point sideways, so the domain is (-\infty, \infty). But every point has height -2, so the range is one number, \{-2\}.

Example 3

Day 1: a drone’s flight

Lazlo flies a drone from a rooftop. Figure 4 shows its height h meters at time t minutes, over the 20 minutes its battery lasts. The graph is made of straight pieces through (0, 10), (5, 50), (9, 50), (13, 20), (16, 30) and (20, 10), with closed dots at both ends. Find the domain and range and say what each means.

Figure 4Example 3. The drone’s height over the 20 minutes its battery lasts.

Solution

The input is the time t and the output is the height h.

Domain. The graph runs from t = 0 to t = 20, both ends closed, with no break:

0 \le t \le 20, \quad\text{that is}\quad [0, 20].

The height is known for every moment of the flight.

Range. A straight piece only passes heights between the heights at its two ends, so the lowest and highest points are among the corners: 10, 50, 50, 20, 30 and 10. The lowest is 10, at the start and the end; the highest is 50, during the hover:

10 \le h \le 50, \quad\text{that is}\quad [10, 50].

The drone was never lower than the 10 m rooftop and never higher than 50 m. The range does not start at 0 just because the axis does, and the dip to 20 m at t = 13 is a low point but not the lowest.

Example 4

Day 2: a graph in two pieces

Find the domain and range of the graph in Figure 5. One piece is the segment from the closed dot (-5, -1) to the closed dot (-2, 2); the other is the segment from the open dot (1, 3) to the closed dot (4, 0).

Figure 5Example 4. The domain has a gap; the two pieces’ heights overlap, so the range does not.

Solution

Domain. The first piece covers -5 \le x \le -2. The second starts just after 1 (the open dot) and ends at 4, included: 1 < x \le 4. Nothing lies above the stretch between -2 and 1, so the domain has a gap and needs \cup:

[-5, -2] \cup (1, 4].

Range. The first piece rises from height -1 to height 2, both reached: [-1, 2]. The second falls from just under 3 to 0: [0, 3). These shadows overlap between 0 and 2, so together they leave no gap. Height 3 is not reached, since the first piece stops at 2. So the range is

-1 \le y < 3, \quad\text{that is}\quad [-1, 3).

The domain has a gap and the range does not.

Example 5

Day 2: a single missing point

Find the domain and range of each graph in Figure 6.

(a) The segment from the closed dot (-4, -2) to the closed dot (4, 2), with the point (2, 1) missing.

(b) The tent from the closed dot (-3, 0) up to (0, 3) and down to the closed dot (3, 0), with the point (1, 2) missing.

Figure 6Example 5. A missing point always removes its input. It removes its output only if no other point has that height.

Solution

(a) Domain. The open dot removes the point above x = 2, and nothing else is in that column, so 2 is lost:

[-4, 2) \cup (2, 4],

that is, -4 \le x \le 4 with x \ne 2.

Range. The segment rises steadily, so each height from -2 to 2 is reached at exactly one point. The only point at height 1 is the missing one, so 1 is lost too:

[-2, 1) \cup (1, 2].

(b) Domain. As before, the hole removes its input: [-3, 1) \cup (1, 3].

Range. The missing point has height 2, but on its way up the tent passes height 2 at (-1, 2), which is on the graph (the dashed line in Figure 6). So 2 is still reached. The lowest height is 0, at both ends, and the highest is 3, at the peak:

0 \le y \le 3, \quad\text{that is}\quad [0, 3].

The hole always cost the domain its input, because a function has only one point above each input. It cost the range its output only in (a), where nothing else reached that height.

Example 6

Day 2: parabolas and their turning points

Find the domain and range of each graph in Figure 7.

(a) The parabola opening upwards with its lowest point at (1, -3) and arrows on both arms.

(b) The parabola opening downwards, drawn only from the closed dot (-4, -5), over its highest point (-1, 4), to the closed dot (1, 0).

Figure 7Example 6. The lowest or highest point of a range can sit at a turning point, not at an end.

Solution

(a) Domain. The arms spread outward forever both ways, so every x is used: (-\infty, \infty).

Range. The lowest point is the vertex, (1, -3), on the graph. From there both arms rise without end:

y \ge -3, \quad\text{that is}\quad [-3, \infty).

Every height above -3 is reached twice, once on each arm, and -3 only once, at the vertex; the range asks only which heights are reached, not how often.

(b) Domain. The graph runs from x = -4 to x = 1, both closed: -4 \le x \le 1, that is [-4, 1].

Range. The highest point is the vertex, height 4, which is not an end. The graph falls away from the vertex on both sides, so the lowest point is the lower of the two ends, at heights -5 and 0. The left arm alone climbs through every height from -5 to 4:

-5 \le y \le 4, \quad\text{that is}\quad [-5, 4].

The right end, at height 0, sits inside the range and adds nothing new.

Example 7

Day 2: a cooling cup of soup

Isadora pours soup at 90^\circ\text{C} into a cup in a room at 20^\circ\text{C}. Figure 8 shows the soup’s temperature T (in degrees Celsius) after t minutes. The graph starts at the closed dot (0, 90) and falls more and more slowly toward the dashed line T = 20: the soup cools toward room temperature and never reaches it, and the arrow says the graph carries on for all later times. Find the domain and range, and compare them.

Figure 8Example 7. The temperature gets as close to 20^\circ\text{C} as you like but never reaches it.

Solution

Domain. The graph starts at t = 0, included, and the arrow carries it to the right forever:

t \ge 0, \quad\text{that is}\quad [0, \infty).

Range. The highest point is the start, T = 90, on the graph. After that the soup only cools. The graph never touches the dashed line, so no point has height 20. But it comes as close to the line as you like, and on its unbroken way down it passes every height between 90 and any height just above 20. So

20 < T \le 90, \quad\text{that is}\quad (20, 90].

The open end at 20 has no open dot on the picture. It comes from the leveling off: the graph approaches height 20 forever and never arrives.

Comparing. The domain goes on without end while the range is a stretch of only 70 degrees, both read from the same curve.

Boundaries

A graph gives only what it shows. If a graph runs off the grid with no arrow, or stops with no dot, its domain and range cannot be read from it.
Readings between grid lines are estimates. An exact end needs the rule.
An unbroken graph passes every height between its lowest and highest points. That is what lets a range be read from two points. A jump must be shown with dots, and each piece read separately.
A context can shrink the domain. Which inputs make sense in a real situation is a reasonable domain for a real situation; the inputs a formula can accept are its implied domain.
The shadows work for any graph. A curve that fails the vertical line test still has a domain and range, read the same way. Only for a function is there exactly one point above each input, which is why the hole in Example 5 always cost the domain its input.

Where this goes next

Reading a graph’s stretch along each axis is the start of reading a graph in pieces of input. The next step, increasing and decreasing intervals, splits the domain into the stretches where the graph rises and the stretches where it falls, and the turning points between them are the maximums and minimums that set the ends of a range. Moving a graph moves its shadows with it: shift a graph up 3 and its range shifts up 3, while its domain stays put. Exponential graphs level off toward a horizontal line exactly like the soup, so their ranges have an open end with no dot. And when a function comes as a rule instead of a picture, its domain is found by asking which inputs the rule can accept, which is the implied domain.

Exercises

Day 1: reading graphs

For Exercises 1 to 6, use the graphs in Figure 9. Write the domain and range of each graph as inequalities and in interval notation.

Figure 9Graphs A to F, for Exercises 1 to 6.

Graph A.
Graph B.
Graph C.
Graph D.
Graph E.
Graph F.

Day 1: from a description

On a second flight, Lazlo’s drone takes off from the ground. Its height h meters at time t minutes is a graph of straight pieces through (0, 0), (6, 45), (12, 45) and (20, 15), with closed dots at both ends. Find the domain and range, and say what each means.

Day 2: reading graphs

For Exercises 8 to 13, use the graphs in Figure 10. Write the domain and range of each graph in interval notation, and as inequalities where that is natural.

Figure 10Graphs G to L, for Exercises 8 to 13.

Graph G.
Graph H.
Graph I.
Graph J.
Graph K.
Graph L.

Day 2: from a description, and comparing

Isadora pours a second bowl of soup. Its temperature graph starts at the closed dot (0, 85) and falls more and more slowly toward the room’s 22^\circ\text{C}, never reaching it, with an arrow to the right. Find the domain and range.
A parabola opens upwards with its lowest point at (-2, -5). Only part of it is drawn: from the closed dot (-4, -1), down through the lowest point, up to the closed dot (1, 4). Find its domain and range.
Among Graphs G to L, which have a domain that is a single interval but a range that is not? Which have a range that is a single interval but a domain that is not?

Reasoning

Sketch two different graphs of functions, each with domain [-2, 3] and range [1, 4]. What does this say about how much a domain and range tell you about a graph?
Hedwig says: “The graph in Figure 1 ends at an open dot at (3, 2), so 2 is not in the range.” Explain what is wrong, and describe when reasoning like hers would be right.
The graph in Example 7 has no open dot, yet its range has an open end. Explain how that can happen, and why no point of that graph has height 20.
A function’s graph has domain [0, 5]. Explain why its range must contain at least one number, and describe a graph whose range is as small as possible. Then explain, with an example from this lesson, why a domain in two pieces does not force the range to be in two pieces.

Answers

Domain [-4, 2], that is -4 \le x \le 2; range [-3, 3], that is -3 \le y \le 3. Both ends of the segment are closed dots.
Domain (-3, 3], that is -3 < x \le 3; range [-2, 4), that is -2 \le y < 4. The open dot (-3, 4) is the left end of the domain and the top of the range, and nothing else reaches 4.
Domain (1, \infty), that is x > 1; range (-3, \infty), that is y > -3. The ray starts just after the open dot (1, -3) and climbs to the right forever.
Domain [-4, 4), that is -4 \le x < 4; range [-3, 4], that is -3 \le y \le 4. The lowest and highest points are the turning points (1, -3) and (-2, 4). The open end at height 2 costs the range nothing: the first piece passes height 2.
Domain (-\infty, \infty) and range (-\infty, \infty). The slanted line runs forever both ways, so it passes every x and every height.
Domain (-\infty, 1], that is x \le 1; range [-3, \infty), that is y \ge -3. The ray starts at the closed dot (1, -3) and climbs to the left forever.
Domain [0, 20], that is 0 \le t \le 20: the height is known for the whole flight. Range [0, 45], that is 0 \le h \le 45: the lowest corner is the take-off at 0 m and the highest the hover at 45 m.
Domain [-4, -1) \cup [1, 4], that is -4 \le x < -1 or 1 \le x \le 4; range [-2, 3], that is -2 \le y \le 3. The pieces give heights (0, 3] and [-2, 1], which overlap.
Domain [-4, 3], that is -4 \le x \le 3; range \{-2, 1\}. Above x = 0 the open dot is missing but the closed dot at height -2 is there, so 0 is in the domain. Each piece is flat, so only two heights are reached.
Domain [-3, -1) \cup (-1, 3], that is -3 \le x \le 3 with x \ne -1; range [-2, 2) \cup (2, 4]. The segment falls steadily, so height 2 is reached nowhere else.
Domain (-\infty, \infty); range (-\infty, 3], that is y \le 3. The highest point is the vertex (1, 3), and both arms fall without end.
Domain [-1, \infty), that is x \ge -1; range [-2, 4), that is -2 \le y < 4. The graph starts at (-1, -2) and rises toward the dashed line at height 4 without reaching it.
Domain [-3, 1) \cup (1, 4]; range (-2, 2], that is -2 < y \le 2. The missing corner is the only point at height -2; the highest point is the left end, at 2.
Domain [0, \infty), that is t \ge 0; range (22, 85], that is 22 < T \le 85. The start, 85, is reached; 22 is approached forever and never reached.
Domain [-4, 1], that is -4 \le x \le 1; range [-5, 4], that is -5 \le y \le 4. The lowest point is the vertex, inside the drawn part; the highest is the higher end, (1, 4).
Graph H has a single-interval domain, [-4, 3], and a range that is not, \{-2, 1\}. Graphs G and L have a single-interval range but a domain in two pieces. Graph I has both in two pieces; J and K have both as single intervals.
For example, the segment from (-2, 1) to (3, 4), and the V from (-2, 4) down to (0, 1) and up to (3, 4). Domain and range record only the two shadows, so infinitely many graphs of different shapes share them. They say nothing about the shape in between.
Height 2 is still reached elsewhere: the piece from (-2, -1) to (1, 4) passes it at x = -0.2. So the range is [-1, 4]. Her reasoning is right only when no other point of the graph has that height, as in Example 1, where the open dot (3, -5) is the only place the segment could be at height -5.
The soup never reaches room temperature: the graph gets closer and closer to the dashed line T = 20 but never touches it, so no point has height 20. Yet it comes within any distance of 20 you like, and the unbroken graph passes every height on the way, so every height just above 20 is reached. The range is the set of heights reached, so its bottom end is 20 with a parenthesis, though nothing on the picture marks it.
The domain is not empty, so the graph has at least one point, for example the one above x = 0, and its height is in the range. The smallest possible range is a single number, as for the flat segment from (0, 3) to (5, 3), whose range is \{3\}. In Example 4 the domain is [-5, -2] \cup (1, 4], but the two pieces’ heights overlap, so the range is the single interval [-1, 3).