HSF.IF.B.5: Relating the Domain of a Function to Its Graph and Context
In plain English: HSF.IF.B.5 is the Common Core functions standard that asks students to connect the domain of a function with its graph and, when the function models a situation, with the quantities it describes. Students decide which inputs make sense, such as whole numbers for counts or a time interval that ends when an event ends, and see this in open and closed endpoints or separate points. It is usually taught in Algebra I.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
Common Core State Standards for Mathematics · Domain: Interpreting Functions (IF) · Cluster: Interpret functions that arise in applications in terms of the context Also written as HSF-IF.B.5 or F-IF.5 · Official standard
The domain of a function is the set of inputs it accepts. This lesson teaches students to see the domain in two places. On a graph, the domain is the set of input values the graph covers: where it starts and stops, whether each end is an open or closed circle, whether there are gaps, and whether the graph is a set of separate points or a connected curve. In a modeling situation, the domain is the set of inputs that make sense for the quantities: a number of engines must be a positive integer, a time runs only until the event ends, and a length cannot be negative.
Students work through the official example (person-hours to assemble n engines, where the positive integers are the appropriate domain) and then compare formulas that accept every real number with models whose context restricts the inputs. They finish by sketching graphs whose shape shows the domain correctly.
Learning Objectives
By the end of this lesson, students will be able to:
Read the domain of a function from its graph, using endpoints, open and closed circles, arrows, gaps and separate points
Choose an appropriate domain for a function that models a situation and justify it with the quantities involved
Decide whether a model's domain is discrete (such as the positive integers) or an interval of real numbers, and draw the graph to match
Explain why a formula's algebraic domain can be larger than the domain that makes sense in context
Prior Knowledge Required
Students should already be comfortable with:
The definition of a function, its domain and its range HSF.IF.A.1
Function notation and evaluating a function HSF.IF.A.2
Writing inequalities for intervals, such as 0 ≤ t ≤ 3 6.EE.B.8
Graphing linear functions and reading points on a graph 8.F.A.3
Write R(n) = 2n on the board and explain: a club sells raffle tickets for $2 each, and 600 tickets were printed. R(n) is the money raised from n tickets.
Warm-Up Prompt
"Which of these inputs make sense for R(n): n = 45, n = 3.5, n = -4, n = 0, n = 750? For each one you reject, give a reason that uses the tickets, not the formula."
Collect answers. The formula accepts every number, but the situation does not: 3.5 tickets cannot be sold, -4 tickets is meaningless, and 750 is more than were printed. 0 and 45 make sense. So the inputs that make sense are the whole numbers from 0 to 600. Tell students that this set is the domain of the model and that today they will learn to see a domain both in a story and on a graph.
Direct Instruction20 minutes
Part 1: Reading the domain from a graph. Project Diagram 2 and give the rules:
Look left and right: the domain is every input value the graph covers, from its leftmost point to its rightmost point.
Check each end: a closed circle means the input is included; an open circle means it is not; an arrow means the graph continues forever in that direction.
Look for gaps: if the graph skips some inputs, the domain skips them too.
Points or curve? If the graph is a set of separate points, the domain is a list of numbers, not an interval.
Write it clearly: in words, as an inequality such as 0 < t ≤ 6, or as a list such as {1, 2, 3, ...}.
Part 2: Choosing a domain from the context. Ask for each model: What does the input count or measure? Can it be a fraction? Can it be negative? When does the situation start and end? Work through the examples below with Diagram 1.
Official example: counting input
h(n) gives the number of person-hours it takes to assemble n engines in a factory. Suppose each engine takes 30 person-hours, so h(n) = 30n.
Equation: Domain: the positive integers 1, 2, 3, ... The graph is separate points (1, 30), (2, 60), (3, 90), ..., not a line.
Interval with both ends included
A car starts with a full 15-gallon tank and uses 1 gallon every 30 miles. The fuel left after d miles is G(d) = 15 - d/30.
Equation: G(d) = 0 when d = 450, so the domain is 0 ≤ d ≤ 450. The graph is a segment from (0, 15) to (450, 0).
Formula domain vs. context domain
A ball is thrown from a 48-foot platform. Its height after t seconds is h(t) = -16t² + 32t + 48.
Equation: The formula works for every real t, but the ball is in the air only from t = 0 until h(t) = 0 at t = 3. Domain: 0 ≤ t ≤ 3.
Open and closed endpoints
A garage charges $4 for the first hour or part of an hour and $2 for each extra hour or part, and the longest stay allowed is 6 hours (Diagram 2).
Equation: Domain: 0 < t ≤ 6. Open circle at t = 0 (no stay, no fee), closed dot at t = 6. F(1) = 4, F(1.5) = 6, F(6) = 14.
Pause on the official example. Discuss why 0 is left out: h(0) = 0 is a true statement, but the standard's example treats "assembling engines" as building at least one, so the positive integers are the natural choice. Students who argue for including 0 should explain what an input of 0 would mean; the key point is that fractions and negatives do not make sense. Also discuss why the engine graph is not connected: 2.5 engines are not assembled.
Guided Practice15 minutes
Pairs use mini whiteboards. Describe four graphs aloud (or sketch them) and have pairs write each domain: (a) a segment from a closed circle at (-2, 5) to an open circle at (4, -1): -2 ≤ x < 4; (b) a line with arrows on both ends: all real numbers; (c) eight separate points at x = 1, 2, ..., 8: {1, 2, ..., 8}; (d) a curve that starts at an open circle at (0, 3) and has an arrow to the right: x > 0. Then give a context: a customer may buy between 1 and 8 concert tickets at $18 each, so C(p) = 18p. Ask which of the four graphs could be the graph of C (graph (c), because p is a whole number from 1 to 8). Listen for students who read the domain from the y-values; ask them which axis shows the inputs.
Independent Practice15 minutes
Students write a domain for each model, justify it in one sentence, and sketch the graph with the correct endpoints: (1) the area A(s) = s² of a square patio with side s meters, for patios with sides up to 5 m (0 < s ≤ 5, a curve with an open circle at the origin and a closed dot at (5, 25)); (2) the temperature T(h) recorded by a weather station h hours after midnight during one day (0 ≤ h ≤ 24, a connected curve); (3) the number of pages P(d) = 15d a student reads in d days, reading every day of a 14-day break (the whole numbers 0 to 14, separate points). Students compare with a partner and discuss any answer where they disagree about whether an endpoint belongs.
Closure5-10 minutes
Exit ticket: A 500 mL bottle of water is full. Each gulp is 25 mL, so W(g) = 500 - 25g is the water left after g gulps. (1) Give the domain. (Answer: the whole numbers 0, 1, 2, ..., 20, since 500/25 = 20.) (2) Should the graph be a line or separate points? Explain. (3) Name one way the domain would look different on the graph if W measured the water left after t seconds of steady pouring.
Differentiation Strategies
For Struggling Students
Give a two-question checklist for every model: "Can the input be a fraction?" and "Where does the situation start and stop?"
Highlight the input axis in one color and have students shade the part of that axis the graph covers before writing the domain
Start with graphs that have integer endpoints and only one open or closed circle
For Advanced Students
Ask for a situation whose domain has a gap, such as a store open from 9 a.m. to 12 p.m. and 1 p.m. to 5 p.m., and a graph that shows it
Compare the algebraic domain of f(x) = 1/(x - 3) with a context such as sharing a prize among x - 3 winners, and explain every restriction
Have students write the domain of the engine model using set-builder notation and explain how it differs from the interval n ≥ 1
Assessment Guidance
What to Look For
Check that students give the domain as input values, not output values: a frequent error is reading the domain from the vertical axis. Look for a reason tied to the quantities, such as "you cannot assemble part of an engine" or "the ball lands at 3 seconds", not only "the graph stops there". On sketches, check that discrete models are drawn as separate points and that open and closed circles match the story. When students include or exclude an endpoint such as 0, accept either choice if the reason fits the context.
02
Classroom Activities
3 Activities
1
Domain Match-Up
20 minGroups of 3
Each group gets 18 cards: 6 situation cards, 6 graph cards and 6 domain cards. Groups match each situation to its graph and its domain and explain every match with the quantities.
Card Set
Total weight of n passengers in an elevator rated for 12 people: domain 0, 1, 2, ..., 12; graph of 13 separate points
Height of a candle that burns down completely in 5 hours: domain 0 ≤ t ≤ 5; graph a segment with two closed ends
Cost of a pizza with k toppings, at most 6 toppings: domain 0, 1, ..., 6; graph of 7 separate points
Area of a circular garden with radius r meters, for radii up to 4 m: domain 0 < r ≤ 4; graph a curve with an open circle at the origin
Cost to mail a letter weighing w ounces, for letters up to 3.5 oz: domain 0 < w ≤ 3.5; graph a step function
Number of bacteria t hours after a lab culture starts, P(t) = 500(2)t, observed for the first 10 hours: domain 0 ≤ t ≤ 10; graph a curve with closed dots at (0, 500) and (10, 512,000)
Procedure
Groups first sort the situations into "counts" and "measurements", then match graphs and domains
For each match, one student writes a sentence that begins "The input is ___, so it can / cannot ___"
Groups check their matches with another group and resolve any disagreement
Discussion Questions
Which situations have a domain that is a list of numbers? What do they have in common?
Why is r = 0 excluded for the garden but t = 0 included for the candle?
The bacteria formula accepts negative t. Why is it left out of the domain here?
2
Same Rule, Different Domains
15-20 minPairs
Pairs graph the same rule, f(x) = 40x, in three situations and see that the context, not the formula, decides the domain and therefore the shape of the graph.
The Three Situations
A caterer earns $40 per hour and is paid by the minute for shifts of up to 8 hours: domain 0 ≤ x ≤ 8, a segment from (0, 0) to (8, 320)
Concert tickets cost $40 each, with 1 to 6 tickets per order: domain 1, 2, ..., 6, six points from (1, 40) to (6, 240)
The rule f(x) = 40x with no context: domain all real numbers, a line with arrows on both ends
Procedure
Each pair draws three separate graphs on the same scale and writes the domain under each
Pairs answer: "What is f(2.5) in each situation, if it exists?" ($100 for the caterer; not defined for tickets; 100 for the pure rule)
Pairs write one new situation for f(x) = 40x with a different domain and trade it with another pair
Modification for Distance Learning
Pairs use a free graphing app, entering the rule with a domain restriction for the first situation and a list of points for the second, and share screenshots of all three graphs.
3
Fix the Graph
20 minGroups of 3-4
Groups receive four graphs drawn by a fictional student, each with a domain error. They find the error, explain it with the context, and redraw the graph correctly on chart paper.
The Four Flawed Graphs
Students on a school bus that holds 48: drawn as a solid line from 0 to 60 (should be separate points for 0, 1, ..., 48)
Height of a stone dropped from a bridge: drawn for negative times and below the water (should start at t = 0 and stop when the stone reaches the water)
Water in a draining 200-liter tank: the line continues below the horizontal axis (should stop when the tank is empty)
Daily profit of a bakery that can make at most 200 loaves: drawn with an arrow to the right (should stop at 200 loaves)
Procedure
Groups write the domain the student used and the domain that fits the context, side by side
Each group redraws one graph on chart paper with correct endpoints and presents it
Challenge Variation
Groups design their own flawed graph for a new situation and trade with another group, which must find and fix the error.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Three Models, Three Kinds of Domain
Left: the official engine example with 30 person-hours per engine, where the domain is the positive integers, so the graph is separate points. Center: fuel left in a 15-gallon tank, a segment with both endpoints included. Right: a ball's height; the formula is defined for every t (dashed), but only 0 ≤ t ≤ 3 describes the ball. All three are drawn to scale.
Diagram 2: Reading the Domain of a Step Function
Parking fee F(t) for a garage that charges $4 for the first hour or part of an hour and $2 for each additional hour or part, with a 6-hour limit, drawn to scale. Each step has an open circle on the left and a closed dot on the right, so the domain is 0 < t ≤ 6.
04
Homework Assignment
~30 min
HSF.IF.B.5 Homework: Domain from Graphs and Context
Directions: Write each domain as an inequality or a list, and justify it in a complete sentence. When you sketch a graph, label the axes with quantities and units, and show open circles, closed dots or separate points where they belong.
Part 1: Domain from a Graph (Problems 1-2)
Give the domain of each graph: (a) a segment from a closed dot at (-3, 2) to a closed dot at (5, 6); (b) a curve that starts at an open circle at (1, -2) and has an arrow to the right; (c) the graph of y = √(x - 3) + 2, which you may draw with a graphing app; (d) five separate points at x = -2, -1, 0, 1 and 2.
A 1,200-gallon pool is drained at a steady 80 gallons per minute. The water left after t minutes is V(t) = 1200 - 80t. Find the domain in context, sketch the graph, and explain what each endpoint of the graph means.
Part 2: Domain from the Context (Problems 3-4)
A club orders custom T-shirts for $9 each plus a $30 setup fee, so C(n) = 9n + 30. The printer requires at least 10 shirts per order and the club can order at most 60. Give the domain, find C(10) and C(60), and explain whether the graph should be a line or separate points.
A stone is dropped from a bridge 78.4 m above a river. Its height after t seconds is h(t) = 78.4 - 4.9t². Find when it reaches the water and state the domain in context. The formula also gives values for t = -1 and t = 5. Explain why neither belongs in the domain.
Part 3: Comparing Domains (Problems 5-6)
A theater has 320 seats and sells every ticket for $15. R(s) = 15s is the revenue from s tickets sold for one show. Give the domain and the range, and explain how you know the domain is not 0 ≤ s ≤ 320 as an interval of all real numbers.
Write two situations for the rule g(x) = 5x: one where the domain is the whole numbers 1, 2, ..., 20, and one where the domain is 0 ≤ x ≤ 3. Sketch both graphs and explain how each graph shows its domain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Domain from Graphs
Correct domains with open and closed ends read correctly
One endpoint or notation error
Domain read from outputs or missing
Domain from Context
Domain fits the quantities and is justified in a sentence
Correct domain, weak or missing reason
Domain does not fit the situation
Discrete or Continuous
Correctly decides points or connected graph and explains why
Correct choice without explanation
Incorrect choice
Sketches
Axes labeled; endpoints and shape match the domain
Shape correct but labels or endpoints missing
Sketch does not show the domain
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Pick an answer for each multiple-choice question to see whether it is right and why. For short-answer questions, write your answer first and then reveal the model answer. Reset quiz clears everything.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
The function m(k) gives the number of minutes it takes a bakery team to decorate the k wedding cakes in an order. Which domain is appropriate for m?
Answer: B
The input counts the cakes in an order, and only whole cakes are decorated, so the positive integers fit. This is the same reasoning as the engine example in the standard. Choice C allows inputs such as 2.7 cakes. Choices A and D allow negative numbers of cakes, which make no sense.
Question 2 of 20 · Multiple Choice
A graph is a segment from a closed dot at (-4, 1) to an open circle at (3, 5). What is its domain?
Answer: A
The graph covers inputs from -4 to 3. The closed dot includes -4 and the open circle excludes 3, so -4 ≤ x < 3. Choice B swaps the open and closed ends. Choice C gives the range, read from the vertical axis. Choice D ignores the open circle.
Question 3 of 20 · Multiple Choice
A graph consists only of the points (0, 3), (2, 7), (4, 11), (6, 15) and (8, 19). What is its domain?
Answer: C
The domain is the set of inputs, the first coordinates of the points: 0, 2, 4, 6 and 8. Choice A includes numbers such as 1.5 that are not on the graph. Choice B lists the outputs. Choice D includes even numbers that do not appear, such as 10 and -2.
Question 4 of 20 · Multiple Choice
The graph of a parabola has its vertex at (1, -3), opens upward, and has arrows on both ends. What is its domain?
Answer: C
The arrows show that the graph continues left and right forever, so every x-value is an input. Choice A is the range, read from the lowest point. Choice B uses only the right half of the parabola.
Question 5 of 20 · Multiple Choice
A graph starts at a closed dot at (2, -1) and rises to the right with an arrow. What is its domain?
Answer: A
The graph covers x = 2 (closed dot, so included) and every larger x (the arrow). Choice B would need an open circle at x = 2. Choice C describes outputs. Choice D ignores that nothing is drawn to the left of x = 2.
Question 6 of 20 · Multiple Choice
T(m) is the temperature of a solution m minutes after the start of a 90-minute lab. Which domain fits the function in context?
Answer: A
The solution has a temperature at every moment from the start (m = 0) to the end of the lab (m = 90), so the domain is 0 ≤ m ≤ 90. If the class only records whole minutes, the data points have inputs 0, 1, ..., 90. Choice C ignores that the lab ends at 90 minutes. Choice D describes the range.
Question 7 of 20 · Multiple Choice
Tickets cost $12 each, and an order must have at least 1 and at most 10 tickets. C(t) = 12t is the cost of an order of t tickets. What is the domain?
Answer: D
The input is a whole number of tickets from 1 to 10. Choice A includes fractions such as 2.5 tickets and an order of 0. Choice B lists the possible costs, which is the range. Choice C has no upper limit.
Question 8 of 20 · Multiple Choice
A(s) = s² gives the area of a square tile with side s cm. Tiles are cut with sides up to 30 cm. How should the graph of A look?
Answer: B
Side lengths are positive and at most 30 cm, and any length in between is possible, so the domain is 0 < s ≤ 30 and the graph is a connected piece of the parabola. A(30) = 900. Choice A includes negative side lengths. Choice C assumes only whole-centimeter sides. Choice D draws the wrong shape: area does not grow linearly with side length.
Question 9 of 20 · Multiple Choice
A car rental company charges by the whole day, so the cost C(d) is defined only for d = 1, 2, 3, ... Why should the graph of C not be a connected line?
Answer: B
A connected line would claim a cost for 2.5 days, but the domain has only whole numbers of days, so the graph is separate points. Choices A and C describe the outputs, which do not decide whether the graph is connected.
Question 10 of 20 · Multiple Choice
A candle is 24 cm tall and burns 1.5 cm per hour. L(t) = 24 - 1.5t is its height after t hours. What is the domain in context?
Answer: B
The candle is gone when 24 - 1.5t = 0, which gives t = 16 hours, so 0 ≤ t ≤ 16. Choice A uses the height as the time limit. Choice C is the range. Choice D would allow negative heights.
Question 11 of 20 · Multiple Choice
A step graph has an open circle at its left end, (0, 5), and a closed dot at its right end, (4, 11). There are no gaps. What is the domain?
Answer: D
The open circle excludes x = 0 and the closed dot includes x = 4, so 0 < x ≤ 4. Choice A includes 0. Choice B leaves out 4. Choice C describes outputs.
Question 12 of 20 · Multiple Choice
Which function has a domain made only of whole numbers?
Answer: C
Songs are counted, so s can only be 0, 1, 2, ... The other inputs are measurements: time and weight can take any value in an interval, such as 2.35 pounds of apples.
Question 13 of 20 · Multiple Choice
A soccer ball is kicked from the ground. Its height in meters after t seconds is h(t) = -5t² + 20t. What is the domain in context?
Answer: D
h(t) = 0 when -5t(t - 4) = 0, so t = 0 (kick) or t = 4 (landing), and the ball is in the air for 0 ≤ t ≤ 4. Choice B stops at the maximum height at t = 2. Choice C is the range, since h(2) = 20.
Question 14 of 20 · Multiple Choice
The side length of a square with area A is s(A) = √A. Which statement is correct?
Answer: A
The square root of a negative number is not real, so the formula needs A ≥ 0; a real square also has positive area, so A > 0 in context. Choice B allows negative areas. Choice D describes the outputs, the side lengths.
Question 15 of 20 · Short Answer
A 900-liter tank starts empty and is filled at 45 liters per minute. V(t) = 45t is the volume after t minutes, until the tank is full. Give the domain and describe the graph.
The tank is full when 45t = 900, so t = 20. The domain is 0 ≤ t ≤ 20 minutes. The graph is a segment from (0, 0) to (20, 900) with both endpoints included, because the model starts at 0 minutes and ends when the tank is full.
Question 16 of 20 · Short Answer
A graph has two pieces: a horizontal segment from a closed dot at (-5, 2) to an open circle at (-1, 2), and a segment from a closed dot at (1, 0) to a closed dot at (6, 5). Write the domain.
The domain is -5 ≤ x < -1 or 1 ≤ x ≤ 6. The inputs from -1 to 1 are not covered, so the domain has a gap. -1 is excluded because of the open circle, and -5, 1 and 6 are included.
Question 17 of 20 · Short Answer
For the official engine example, a student says the domain of h(n) should be 0 ≤ n ≤ 50, because the factory can assemble at most 50 engines a month. What is right about this answer, and what should change?
The upper limit is a good use of the context: if h is used for one month of production, n cannot be more than 50. But the interval 0 ≤ n ≤ 50 includes fractions such as 12.5 engines. The domain should be the integers 1, 2, ..., 50, matching the official example's choice of positive integers, and the graph should be separate points.
Question 18 of 20 · Short Answer
A rectangle has a perimeter of 40 m. Its area in terms of its width w is A(w) = w(20 - w). What is the domain in context, and how does the graph of the model differ from the graph of the formula?
The width and the length 20 - w must both be positive, so the domain is 0 < w < 20. The formula's graph is a whole parabola, but the model uses only the arch above the w-axis between w = 0 and w = 20, with open circles at (0, 0) and (20, 0). The largest area is A(10) = 100 m².
Question 19 of 20 · Short Answer
A bike share charges $3 for each 30 minutes or part of 30 minutes, and a ride can last at most 2 hours. Give the domain of the cost function C(t), with t in minutes, and describe the ends of each step on the graph.
The domain is 0 < t ≤ 120 minutes. The costs are $3 for 0 < t ≤ 30, $6 for 30 < t ≤ 60, $9 for 60 < t ≤ 90 and $12 for 90 < t ≤ 120. Each step has an open circle on the left and a closed dot on the right, because a ride of exactly 30 minutes costs $3, not $6.
Question 20 of 20 · Short Answer
A town's population t years after 2020 is modeled by P(t) = 1500(1.03)t. The formula works for every real t. Why might the domain be restricted, and what domain would you choose?
P(0) = 1500 is the population in 2020. Negative t describes years before 2020, where the data behind the model may not apply, and very large t assumes 3% growth forever. A reasonable choice is t ≥ 0, limited to the years the model is meant to predict, such as 0 ≤ t ≤ 10. If the population is counted once a year, the inputs are the whole numbers in that interval.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.IF.B.5 mean?
HSF.IF.B.5 means students can connect the domain of a function to its graph and, when the function models a real situation, to the quantities it describes. On a graph they read which inputs are covered; in context they decide which inputs make sense, such as positive integers for a number of engines or 0 ≤ t ≤ 3 for the time a ball is in the air.
Is HSF.IF.B.5 taught in Algebra 1?
Yes. Many course sequences teach it in Algebra I alongside linear, quadratic and exponential models, and use it again in Algebra II whenever a model is built for a situation. It follows naturally after HSF.IF.A.1, which defines domain and range.
What is the official example in HSF.IF.B.5?
The standard says that if h(n) gives the number of person-hours it takes to assemble n engines in a factory, the positive integers would be an appropriate domain. The input counts whole engines, so fractions and negative numbers do not make sense, and the graph is a set of separate points rather than a line.
What is the difference between the domain of a formula and the domain in context?
The formula's domain is every input the expression can accept. The domain in context is only the inputs that make sense for the situation. h(t) = -16t² + 32t + 48 can be evaluated for any t, but a thrown ball is in the air only from t = 0 until it lands at t = 3, so the model's domain is 0 ≤ t ≤ 3.
How do you find the domain from a graph?
Look along the horizontal axis. Find the leftmost and rightmost inputs the graph covers, check whether each end is a closed dot (included), an open circle (excluded) or an arrow (continues forever), and look for gaps. If the graph is separate points, list their inputs.
When should a graph be separate points instead of a line?
When the input counts things that cannot be split, such as engines, tickets, people or books. A line would claim outputs for inputs like 2.5 tickets. When the input is a measurement such as time, distance or weight, the graph is usually connected over an interval.
What are common mistakes with domain?
A common error is reading the domain from the vertical axis, which gives the range instead. Others are ignoring open circles, forgetting that a situation ends (a tank empties, a ball lands), and connecting points for a counting input. Students also sometimes include negative times or lengths because the formula allows them.
Should 0 be in the domain?
It depends on the situation, and students should explain their choice. A tank at t = 0 minutes is a real starting moment, so 0 belongs. A circular garden with radius 0 is not a real garden, so 0 is left out. For the engine example the standard uses the positive integers, which leave out 0. Accept either choice when the reason fits the context.
How do students write a domain?
In words ("the whole numbers from 0 to 20"), as an inequality (0 < t ≤ 6), as a list ({1, 2, 3, ..., 10}) or in interval notation ((0, 6]). In Algebra I, words and inequalities are usually enough; interval notation is often added in Algebra II and Precalculus.
How does HSF.IF.B.5 connect to other standards?
It builds on HSF.IF.A.1 (domain and range) and works together with HSF.IF.B.4, since key features only matter on the domain where the model applies. It also supports HSA.CED.A.3, where students decide whether solutions are viable in a context, and it is used when students write inverse functions and need to restrict a domain (HSF.BF.B.4).
07
Related Standards
6 standards
These standards connect to HSF.IF.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.IF.A.1Prerequisite
Understand a function as assigning each domain element exactly one range element
