math-2017:8.F.1
math-2017:8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.
Learning goal: Explain what makes a rule a function, decide from a table, a list of ordered pairs or a graph whether a relationship is a function, and describe the graph of a function as the set of ordered pairs made from each input and its corresponding output.
What a function is. A function is a rule that assigns to each input exactly one output. There are two conditions in that sentence. First, every input the rule accepts gets an output. Second, no input gets more than one output. The rule "double the input and add 1" is a function: input 3 gives 2 × 3 + 1 = 7 and nothing else. Input 0 gives 1, input 10 gives 21 and input −4 gives −7. Whatever number goes in, exactly one number comes out, and the same input always produces the same output.
A rule can be given in words, as an equation, as a table or as a graph. Written as an equation with y for the output and x for the input, the rule above is y = 2x + 1. In Grade 8 a function is written with words or with an equation in x and y; the special function notation used in later courses is not required here. What matters is the idea: one input in, one output out.
Exactly one output, not exactly one input. The definition says nothing about whether two different inputs may share an output. They may. The rule "take the absolute value of the input" sends both −2 and 2 to the output 2. That is still a function, because each input, taken on its own, has exactly one output. The rule y = 5, which sends every input to 5, is also a function, since each input still gets one output. Students often reverse the condition and reject these rules. The test is always whether an input has more than one output, never whether an output has more than one input.
A rule that is not a function. Consider the rule "the output is a number whose square is the input." Input 9 could be sent to 3 or to −3, because both square to 9. One input with two candidate outputs means this rule is not a function. Compare it with "the output is the square of the input," y = x², which is a function: input 3 gives 9, input −3 also gives 9, and every input gives exactly one value.
Deciding from a table. A table lists inputs in one column and outputs in another. Read down the input column and look for a repeated input. If an input appears twice with different outputs, the table is not a function. Suppose input 2 appears in two rows, once with output 6 and once with output 8. Input 2 would have two outputs, so this is not a function. If a repeated input has the same output in both rows, the row is only repeated information and the table can still describe a function.
The graph is a set of ordered pairs. Every input together with its corresponding output makes an ordered pair (input, output). The input is written first. The graph of a function is the set of all these ordered pairs, plotted with the input on the horizontal axis and the output on the vertical axis. For y = 2x + 1 with the inputs 0, 1, 2, 3 and 4, the ordered pairs are (0,1), (1,3), (2,5), (3,7) and (4,9). Those five points are the graph when the inputs are limited to those five numbers. If the rule accepts every number as an input, the graph has infinitely many points, and for this rule they lie on one straight line.
Order matters. The pair (2,5) means input 2, output 5. The pair (5,2) would mean input 5, output 2, which is false for this rule, because 2 × 5 + 1 = 11. Swapping the coordinates plots a different point and describes a different relationship.
Deciding from a graph. Because the input is the first coordinate, two points with the same first coordinate represent the same input. If a graph contains (6,2) and (6,7), the input 6 has two outputs, 2 and 7, so the graph cannot be the graph of a function. Picture a vertical line drawn through x = 6: it would pass through both points. That is why points stacked one above the other fail the definition. A circle is a familiar example. The circle x² + y² = 25 contains both (3,4) and (3,−4), because 9 + 16 = 25 either way, so the input 3 has two outputs and the circle is not the graph of a function. You do not need to memorize a rule about vertical lines; you only need to ask whether any input appears with two outputs.
A fully worked model. Take the rule "the output is the input squared, minus 1," so y = x² − 1, with inputs −3, −1, 0, 2 and 3. Compute each output: (−3)² − 1 = 9 − 1 = 8; (−1)² − 1 = 1 − 1 = 0; 0² − 1 = −1; 2² − 1 = 3; 3² − 1 = 8. The ordered pairs are (−3,8), (−1,0), (0,−1), (2,3) and (3,8). Check the definition: each input appears once, with one output, so the rule is a function. Inputs −3 and 3 share the output 8, which is allowed. Plot the five points; no two of them share a first coordinate, which is the graphical way of seeing that each input has exactly one output.
Why the definition matters. In the rest of this unit you will compare functions, decide whether they are linear and build them from situations. Every one of those tasks assumes that a given input produces one predictable output. A rule that could give two answers for the same input would make phrases such as "the rate of change" or "the value at x = 4" meaningless. The definition is the foundation, not a technicality.
Worked table decision: A table has inputs 1, 2, 3, 2 with outputs 4, 6, 8, 6. Input 2 appears twice, but both times its output is 6, so no input has two different outputs and the table describes a function. Change the last output to 7 and the table no longer describes a function: input 2 would be paired with 6 and with 7. The check is a search for a repeated input with different outputs, not a search for repeated outputs.
Worked ordered-pair decision: Set A is {(1,2), (2,3), (3,5), (1,4)}. The first coordinates are 1, 2, 3 and 1; input 1 appears with outputs 2 and 4, so set A is not the graph of a function. Set B is {(1,2), (2,2), (3,2)}. Every first coordinate is different, so each input has exactly one output; set B is the graph of a function even though all three outputs are the same number, 2.
Worked graph of a rule: For y = 3x − 2 with inputs 0, 1, 2 and 3, the outputs are 3 × 0 − 2 = −2, 3 × 1 − 2 = 1, 3 × 2 − 2 = 4 and 3 × 3 − 2 = 7, giving the ordered pairs (0,−2), (1,1), (2,4) and (3,7). Plot each with the input horizontally and the output vertically. The points rise by 3 for each step of 1 to the right, and no two share a first coordinate. If every number were allowed as an input, the graph would be the whole straight line through these points.
Representation task: For the rule y = 2x + 1, make a three-column record for the inputs −2, −1, 0, 1, 2, 3 and 4: the input, the computed output and the ordered pair. Plot the seven ordered pairs on labeled axes. Then add the two points (6,2) and (6,7) in a different color and write one sentence explaining why the graph with those two points added is not the graph of a function, using the words input and output.
Independent performance: Write three rules of your own: one given in words, one as an equation and one as a table of at least five rows. Exactly one of the three must fail to be a function, and it must fail because of a repeated input with different outputs, not because of a repeated output. For each rule, state whether it is a function and justify the decision in one or two sentences. For the two functions, list their ordered pairs for at least four inputs and plot them. Success criteria: every output is computed correctly; the non-function is identified for the right reason; ordered pairs are written input first; and the justification names the input that has two outputs where that happens.
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
Is the rule "the output is 4 times the input, plus 3" a function? Give the outputs for the inputs −1, 0 and 2.
Yes. Each input is sent to exactly one output, because multiplying by 4 and adding 3 gives a single result. The outputs are 4 × (−1) + 3 = −1, 4 × 0 + 3 = 3 and 4 × 2 + 3 = 11, so the ordered pairs are (−1,−1), (0,3) and (2,11).
A table has inputs 4, 5, 6, 4 with outputs 10, 12, 14, 11. Does the table describe a function? Explain.
No. The input 4 appears twice with different outputs, 10 and 11. A function must assign exactly one output to each input, so an input with two outputs breaks the definition. The repeated input, not any repeated output, is what decides this.
The rule "the output is the absolute value of the input" sends −5 and 5 both to 5. Is it a function? Explain.
Yes. The definition requires each input to have exactly one output; it does not forbid two inputs from sharing an output. Input −5 has the single output 5 and input 5 has the single output 5, so the rule is a function.
Which set of ordered pairs is the graph of a function: set A = {(1,3), (2,3), (3,3)} or set B = {(1,3), (1,4), (2,5)}? Explain.
Set A. Its first coordinates 1, 2 and 3 are all different, so each input has one output (all three happen to be 3, which is allowed). In set B the input 1 appears with outputs 3 and 4, so set B is not the graph of a function.
Why is the circle x² + y² = 100 not the graph of a function?
The circle contains both (6,8) and (6,−8), because 6² + 8² = 36 + 64 = 100 and 6² + (−8)² = 100 as well. The input 6 therefore has two outputs, 8 and −8. Since a function assigns exactly one output to each input, the circle cannot be the graph of a function. (Any input between −10 and 10 other than ±10 shows the same thing.)
The graph of a function contains the point (2,7). Can it also contain (2,9)? Can it also contain (5,7)? Explain each answer.
It cannot contain (2,9), because the input 2 would then have two outputs, 7 and 9. It can contain (5,7), because that point has a different input, 5, and two different inputs are allowed to share the output 7.
Write the ordered pairs of y = x² for the inputs −2, −1, 0, 1 and 2, and state whether the rule is a function.
The outputs are 4, 1, 0, 1 and 4, so the ordered pairs are (−2,4), (−1,1), (0,0), (1,1) and (2,4). It is a function because each input has exactly one output; the shared outputs 4 and 1 come from different inputs, which the definition allows.