Before you begin
This quiz finishes family 14. Lines on the Grid already teaches most of it: where a point lives on the grid, what f(x) means, how to work out f(5), and how to find the input that gives a stated output. If you have not been through that, go there first. If the grid itself is new, start one step earlier, with The Coordinate Plane.
Two things are left, and they are what this quiz is for. The first is deciding whether a relationship is a function at all. The second is matching: given a table or a graph, finding the rule that produced it, or picking it out from a list of rules.
Neither needs much arithmetic. Both need you to check carefully rather than guess, and both have one idea that, once you have it, makes most of the questions short.
What a function is
A function is a rule that gives exactly one output for each input.
A vending machine is the plainest picture of one. You press B4, and you get whatever is in B4. Every time you press B4, you get the same thing. If B4 sometimes gave you chips and sometimes a can of soda, you could not use the machine; you would never know what pressing a button meant. A function is a rule you can rely on in exactly that way.
The rules in the rest of family 14 are all functions. f(x) = 2x + 1 takes 3 and gives 7, every time. It never gives 7 on Monday and 8 on Tuesday.
The one thing that is not allowed
There is exactly one way for a relationship to fail to be a function: one input giving two different outputs.
What is allowed surprises people, so it is worth saying just as clearly: two inputs can share an output. In the vending machine, B4 and B5 could both hold the same brand of chips. That is fine; each button still gives one thing. What would break it is one button giving two things.
So when you check whether something is a function, you are looking for one thing only: an input that turns up twice with different outputs. Repeated outputs do not matter at all.
Checking a table or a list of pairs
Relationships are often given as a table, or as a list of pairs written (input, output). Look down the input column for any number that appears more than once. If none repeats, it is a function. If one does, look at its outputs: if they are the same, that is fine; if they differ, it is not a function.
| input x | 1 | 2 | 3 | 2 |
|---|---|---|---|---|
| output y | 4 | 6 | 8 | 5 |
The input 2 appears twice, once with 6 and once with 5. One input, two outputs: not a function.
| input x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| output y | 6 | 6 | 8 | 8 |
No input repeats, so this is a function. The outputs 6 and 8 each appear twice, and that does not matter.
Mapping diagrams
A mapping diagram draws the same thing with arrows: inputs in one oval, outputs in the other, and an arrow from each input to its output. The check becomes a matter of counting arrows.
Every input must have exactly one arrow leaving it. Two arrows arriving at the same output is fine. Two arrows leaving the same input is not.
On a graph: the vertical line test
On a graph the input is along the bottom and the output is up the side. A single input is a single position along the bottom, and everything directly above or below it is an output for that input. So: draw a vertical line through the graph, anywhere. If it can ever cross the graph more than once, it is not a function.
On the left, wherever you put the vertical line, it meets the curve once, so each input has one output. On the right, the line meets the circle twice: the input 1.6 has one output above the axis and another below it. A circle is never a function, and neither is any shape that turns back on itself sideways.
Matching a table to its rule: check every row
The most common matching question gives you a table and several rules, and asks which rule made the table. The method is simple and has one trap.
The method: put each input into each rule and see whether the output comes out right.
The trap: checking only the first row. One row fits a great many rules. Here is a table and three candidates:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 1 | 3 | 5 | 7 |
| y = x + 1 | 1 | 2 | 3 | 4 |
| y = x² + 1 | 1 | 2 | 5 | 10 |
| y = 2x + 1 | 1 | 3 | 5 | 7 |
All three rules get the first row right. Only y = 2x + 1 gets every row right, so that is the rule. y = x² + 1 even gets a second row right by accident, which is exactly why one or two rows prove nothing. A rule matches a table only if it matches every row.
Finding the rule when it is a straight line
Sometimes there are no candidates; you are asked to find the rule yourself. When the table comes from a straight line, there is a quick way, and it rests on one thing to look at: how much y changes each time x goes up by 1.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | −2 | 1 | 4 | 7 | 10 |
| change | +3 | +3 | +3 | +3 |
Every time x goes up by 1, y goes up by 3. That steady change is the slope, the m in y = mx + b. And the output when x is 0 is the starting value, the b: here, −2. So the rule is y = 3x − 2.
Then check it on a row you did not use: when x is 4, 3 × 4 − 2 = 10. It matches.
Two things can make this harder, and both are easy to deal with.
The x values go up by more than 1. If x goes 0, 2, 4, 6 and y goes up by 6 each time, then y goes up by 6 for every 2 of x, which is 3 for every 1. Divide the change in y by the change in x. That is what slope has always meant: rise over run.
There is no row for x = 0. Then work backward. If x = 1 gives 5 and the slope is 3, then x = 0 must give 5 − 3 = 2, and that is b.
When the changes are not the same
If the change in y is not steady — up by 3, then by 5, then by 7 — the table does not come from a straight line, and y = mx + b cannot describe it. Something else is going on, often a square. The table 1, 4, 9, 16 goes up by 3, 5, 7: those are the squares, y = x².
This is worth knowing for its own sake, because a question will sometimes simply ask whether a table is linear. Linear means a straight line, and a straight line means the changes are steady. Getting bigger is not enough; it has to get bigger by the same amount each step.
Matching a graph to its rule
For a straight-line graph, read the rule off in two steps.
First, where does it cross the vertical axis? That is b. Here it is 1.
Second, what is the slope? Find two points where the line passes exactly through a corner of the grid — here (0, 1) and (2, 5). Count how far across and how far up from one to the other: 2 across and 4 up. Up divided by across is 4 ÷ 2 = 2. If the line goes down from left to right, the slope is negative.
So the rule is y = 2x + 1. Lines on the Grid teaches slope in much more detail if this is new.
When the question gives you a list of rules to choose from, there is an even faster way: take a point you can read exactly off the graph and try it in each rule. Any rule that does not produce that point is out. Usually one point leaves one survivor. If two survive, try a second point.
The guard
Not a function means one input with two outputs. Nothing else. Repeated outputs are fine.
The test on a graph is a vertical line, because a vertical line is one input.
Check every row of a table, not the first one. The wrong answers in a matching question are chosen to fit the first row.
Steady change means a straight line. Divide the change in y by the change in x for the slope, and find b where x is 0.
Try a point. Any rule that fails one point from the graph or one row of the table is wrong, however good it looked.
Three to study before you start
Which of these are functions? (a) the pairs (1, 3), (2, 5), (3, 5), (4, 9). (b) the pairs (5, 1), (6, 2), (5, 3), (7, 4).
Look only at the first number of each pair, the input, and ask whether any repeats.
(a) The inputs are 1, 2, 3, 4. None repeats. It is a function. The output 5 appears twice, for 2 and 3, and that is allowed.
(b) The input 5 appears twice, once with the output 1 and once with the output 3. One input, two outputs. It is not a function.
Notice that you never had to look for a rule. Whether something is a function is only a question of whether any input has two outputs.
Find the rule for this table: x is 1, 2, 3, 4 and y is 5, 9, 13, 17.
The changes: 9 − 5 = 4, 13 − 9 = 4, 17 − 13 = 4. Steady, so it is a straight line, and since x goes up by 1 each time, the slope is 4.
There is no row for x = 0, so work backward one step: when x is 1, y is 5, so when x is 0, y is 5 − 4 = 1. That is b.
The rule is y = 4x + 1. Check it on the last row: 4 × 4 + 1 = 17. It matches.
A straight line passes through (0, −3) and (2, 1). Which rule is it: y = 2x − 3, y = −3x + 2, or y = 4x − 7?
Try the point (2, 1) in each. 2 × 2 − 3 = 1, which works. −3 × 2 + 2 = −4, which does not, so that rule is out. 4 × 2 − 7 = 1, which also works.
Two rules survived the first point, which happens. Try the second point, (0, −3). 2 × 0 − 3 = −3, which works. 4 × 0 − 7 = −7, which does not.
So the rule is y = 2x − 3. One point narrowed it down; the second settled it. That is the same lesson as the table: one agreement proves nothing, and you keep checking until only one is left.
Now you
Work on paper. No calculator is needed. For every matching question, check your rule against every row or every point you can read, not just one. Questions 6 and 10 have two parts.
- Is this relationship a function? Say why or why not. (2, 5), (3, 7), (2, 9), (4, 11)
- Is this relationship a function? Say why or why not. (1, 4), (2, 4), (3, 4), (4, 4)
- Find the rule for this table.
x 0 1 2 3 y 3 7 11 15 - Which rule produced this table?
x 1 2 3 4 y 2 5 10 17 - A) y = 3x − 1
- B) y = x² + 1
- C) y = 2x + 1
- D) y = x + 1
- Write the rule for the line in this graph.
- Use the four graphs below.
(a) Which of them are not functions?
(b) For one of them, say how you know.
Reading A reading rest stop, for the stubborn ones.
Ten minutes with the last section of this page — The Company, below the answer key — before you finish the quiz. It is about a New York office in the Depression where hundreds of people who had been out of work were hired to calculate tables of functions by hand, and how they checked their own work with the same steady-change idea you have just learned. It will not help you with question 7. It may change what you picture when you hear the word computer.
No photograph at this rest stop. The graphs in this quiz are all above.
- Find the rule for this table.
x 0 2 4 6 y 5 11 17 23 - Keisha looks at this table and says it must be a straight line, because y gets bigger every time x does. Is she right? Explain, and find the rule.
x 1 2 3 4 y 1 4 9 16 - Which rule matches this graph?
- A) y = 2x + 4
- B) y = −2x + 4
- C) y = −½x + 4
- D) y = 4x − 2
- A phone plan’s monthly cost depends on the minutes used, as in this table.
(a) Write a rule for the cost, C, in terms of the minutes, m. (b) What would 450 minutes cost?minutes 0 100 200 300 cost ($) 20 28 36 44
Check your work
When computers were people
In 1938, in the tenth year of the Depression, the federal government opened an office in New York City to make tables.
The tables were of functions: long printed lists giving the value of some rule for thousands of inputs, one row after another, to many decimal places. Engineers, navigators, physicists and statisticians needed them the way a carpenter needs a ruler. Before electronic computers, if you needed the value of a function you did not calculate it; you looked it up in a book of tables. The trouble was that many of the tables had never been made, and the ones that had were full of errors.
The office was called the Mathematical Tables Project, and it was paid for by the Works Progress Administration, the New Deal program that put unemployed people to work on public projects — roads, parks, murals, guidebooks, and, here, arithmetic. At its height it employed several hundred people. Most were not mathematicians. They had been clerks, salesmen, bookkeepers, people who had lost their jobs and taken the one that was offered. The job title was computer. A computer was a person who computed.
The work was organized by two mathematicians. Arnold Lowan directed the project. Gertrude Blanch planned the mathematics. She had come to America from Poland as a child, worked for years as an office clerk, and earned her doctorate at Cornell in 1935, only to find no university job in the Depression. What she did at the project was ingenious and humane at once: she broke each hard calculation down into long chains of simple steps, so that people who could add and subtract accurately, and nothing more, could produce values that were right to many decimal places. The difficult part was in the planning. The labor was spread across many hands.
And they checked the results with the idea in the middle of this quiz.
When a function is smooth, its values in a table change steadily. You saw the simplest case above: a straight line changes by the same amount every step. A curve does not change by the same amount, but the changes in the changes are steady — the table 1, 4, 9, 16 goes up by 3, 5, 7, and those go up by 2, 2, 2 — and for the smooth functions the project worked on, taking changes of changes a few times over settles into a column of small, steady numbers. A copying mistake, a slipped digit, a wrong subtraction, breaks that smoothness. It shows up as a sudden jump in a column of differences that should have been calm. So the finished tables were checked by differencing, row by row, and an error announced itself.
The project published volume after volume of tables. When the WPA ended, the work was taken over by the National Bureau of Standards, and it led in the end to a single reference book, the Handbook of Mathematical Functions, published in 1964 and used by scientists and engineers for decades afterward. Electronic computers made the human ones unnecessary within a generation. The tables they made went on being used long after.
Source: David Alan Grier, When Computers Were Human (Princeton University Press, 2005), which tells the history of the Mathematical Tables Project in detail.