At a glance. One lesson of about 60 minutes. It assumes function machines from Year 7 and graphs of straight lines from Year 9. Students decide whether a relation is a function from pairs, graphs and equations, state a domain and range, and evaluate f(x).
Calculator: GDC optional. Everything can be done by hand. The optional section shows students with a GDC how a table of values checks f(x), and warns that a graph window can hide the domain.
What students will learn
Know
- A function gives each element of the domain exactly one element of the range.
- The vertical line test, and the notation f(x).
- The domain is the set of inputs; the range is the set of outputs reached.
Understand
- Two inputs may share an output, but one input may never have two outputs.
- A graph, a table, a set of pairs and an equation can all represent the same relation.
- The domain is part of the definition of a function.
Do
- Decide whether a relation is a function from any representation.
- State the domain and range of a function.
- Evaluate f(a) and solve f(x) = k for x.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: evaluating functions and solving f(x) = k (questions 2 to 7).
- Criterion B. Investigating patterns: the what-if. Students test rules for whether each output comes from one input and generalize.
- Criterion C. Communicating: giving a reason in words for why a relation fails the vertical line test (the check and the exit ticket).
- Criterion D. Applying mathematics in real-life contexts: the taxi fare, a function used as a model and then run backwards (question 5).
The one thing to do with the figure
Ask for a prediction before every slide. With x = y² chosen, ask: “How many outputs will the line x = 1 meet?” before anyone moves the slider. Then slide to x = 0 and ask why there is only one.
Then press y = √x and slide the line to the left of the graph. Ask what the domain is, and how the figure shows it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: cards and borrowers. Collect predictions for both directions. |
| 12 min | The figure with predictions; the key idea; settle the hook. |
| 10 min | Warning card and the worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, C or D, return to the figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 13; y = 2x − 1; 43. |
|---|---|
| Hook | Card to borrower is a function (each card has one borrower). Borrower to card is not, if a borrower holds two cards. |
| Worked examples | Range {2, 3, 7}; f(−2) = 15; domain x ≥ 1 and range g(x) ≥ 0. |
| Faded example | 1. |
| 1. Recognizing a function from ordered pairs | A. B, C and D each repeat an input with different outputs; A repeats only the output. |
| 2. Evaluating a function at a negative input | 28. 9 + 12 + 7. |
| 3. Finding the input for a given output | −3. −2x = 6. |
| 4. Counting the values in a range | 3. Range {−1, 0, 3}. |
| 5. Using a function as a model | 9.5. 4d = 38. |
| 6. Domain of a square root | 4. x ≥ 4. |
| 7. Finding where a function is zero | 6. x(6 − x) = 0. |
| 8. Which relation is not a function | C. x = 5 gives y = 2 and y = −2. A, B and D give one y for each x. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Thinks a repeated input is allowed, or looks at the outputs instead of the inputs. | Underline the first numbers only. Does any first number appear twice with different second numbers? |
| 10 or 4 in question 2 | Squared without brackets, or slipped on the sign of the middle term. | Write (−3) in a bracket every time x appears. |
| 3 or −17 in question 3 | Sign slip when dividing by −2, or put 11 in as the input. | Ask: is 11 the input or the output here? |
| 5 or 2 in question 4 | Counted inputs, or counted the repeated values. | Write the outputs, cross out repeats, then count what is left. |
| 15.5, 12.5 or 38 in question 5 | Added the fixed charge, ignored it, or forgot to divide. | Draw the fare as 12 plus four per kilometre and undo the steps in reverse. |
| 9, −4 or 0 in question 6 | Subtracted instead of dividing, moved 12 with the wrong sign, or tried 0. | Test x = 4 and x = 3 in the root: which is allowed? |
| 3, −6 or 36 in question 7 | Used the turning point, a sign slip, or squared. | Factorize first: x(6 − x). When is a product zero? |
| A, B or D in question 8 | Thinks a curve or a square root cannot be a function, or does not test an input. | Choose x = 5 and find every y in each equation. |
Questions worth asking
- Why can two inputs share an output but one input not have two outputs? Listen for “a function is a machine: the same input must always give the same output”.
- Is the circle x² + y² = 16 a function? What if we keep only the top half? The top half, y = √(16 − x²), passes the vertical line test: choosing a half restricts the range so each x has one y.
- Does the domain belong to the function? Yes: y = x² for all x and y = x² for x ≥ 0 have different graphs and different ranges.
- How is f(x) different from f × x? f is the name of the rule; f(x) is the output of the rule for the input x.
Differentiation
A way in for anyone stuck
- Start with a function machine drawing: input, rule, output, and a mapping diagram with arrows before any graph.
- Give a table with three columns, input, rule and output, and have students fill it for each x before writing f(x).
- For negative inputs, have students write the bracket first and only then simplify.
Stretch
- Find a restriction of the domain of y = x² − 4 that makes it one-to-one, and state the range.
- A relation is given by x² + y² = 16. Write two functions whose graphs together make up the circle.
Students learning in English as an additional language
“Function” here means a rule with one answer for each input; it is not the same as “job” or “purpose”. “Domain” and “range” are everyday words with a special meaning. Say “f of x” for f(x).
| Word | Plain meaning | Example |
|---|---|---|
| function | a rule that gives every input exactly one output | the cost of a parcel from its weight |
| domain | the set of inputs a function can accept | temperatures from 0 to 40 |
| range | the set of outputs a function actually gives | fares from 8 up to 90 |
| relation | any set of ordered pairs, whether or not it is a function | {(1, 2), (1, 3)} |
| vertical line test | a graph is a function if no vertical line meets it twice | a circle fails it |
| f(x) | the output of the function f for the input x | f(2) means put 2 in place of x |
Sentence starters for explaining aloud or in writing:
- This is a function because each input …
- This is not a function because the input … has outputs … and …
- The domain is … and the range is …
Two things not to say
“It is a function if it is an equation.” Some equations, such as x = y², are not functions; test an input.
“Never repeat a number.” Outputs may repeat; only a repeated input with different outputs breaks the rule.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Evaluatingf(x) = 4x − 9. Find f(5).
11. - PairsIs {(1, 2), (2, 2), (3, 2)} a function?
Yes: the inputs are different.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA parking meter in Kyoto charges P(h) = 150 + 90h yen for h hours. A driver pays 510 yen. For how many hours?
150 + 90h = 510, so 90h = 360 and h = 4 hours. - Earlier work (Year 9)Find the gradient of the line through (0, 3) and (4, 11).
(11 − 3) ÷ 4 = 2.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningg(x) = 1 ÷ (x − 2). Which input is not in its domain, and why?
x = 2, because it makes the denominator 0. - RangeState the range of f(x) = x² + 3.
f(x) ≥ 3, because x² is never negative.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider or a button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
The With a GDC section is closed by default and optional. Students without a GDC lose nothing; every answer is reached by hand.