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8th Grade · 8.F.A.1

What is a function in 8th grade

What is a function? In 8th grade math, a function is a rule that takes an input and gives back exactly one output. Your child learns to look at a table, a list of number pairs or a graph and decide whether it shows a function. Your child also reads the output for any input.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.F.A.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.

What it means, in plain words

The short answer to "what is a function" is a rule that turns each input into exactly one output. Input and output are the words for what goes in and what comes out. The input is the number you put in, and the output is the number you get back. The rule "add 3" is a function: an input of 5 always gives an output of 8.

The key word is "exactly". One input can never give two different outputs. Two inputs may share an output, though. The rule "square the number" sends both 3 and −3-3 to 9, and it is still a function. In 8th grade, functions come as equations, tables, mapping diagrams (arrows from each input to its output) and lists of ordered pairs. An ordered pair, such as (2,5)(2, 5), is a number pair that lists the input first and its output second.

The graph of a function is all of its ordered pairs drawn as points. Inputs go along the horizontal axis and outputs go up the vertical axis. Because each input has one output, a function never has two points straight above each other. Your child does not need the f(x)f(x) notation yet. It comes in high school.

Your child should be able to:

  • Explain that a function gives exactly one output for each input
  • Decide whether a table, a mapping diagram or a list of ordered pairs shows a function
  • Read the output for a given input from a table or a graph
  • Explain why two points straight above each other on a graph cannot belong to a function

Example

Do the ordered pairs (1, 4), (2, 6), (3, 6) and (1, 9) show a function?

  1. Look for an input that appears more than once.
  2. The input 1 appears twice: once with the output 4 and once with the output 9.
  3. The output 6 also appears twice, but it comes from two different inputs, 2 and 3. That is allowed.
  4. The input 1 has two outputs, so these pairs do not show a function.

Example

A function's rule is "multiply by 2, then add 1". What is the output when the input is 4?

  1. Multiply the input by 2: 4×2=84 \times 2 = 8.
  2. Add 1: 8+1=98 + 1 = 9.
  3. The output is 9, so the ordered pair is (4,9)(4, 9).

Where children usually get stuck

  • Your child says the pairs (2,5)(2, 5) and (3,5)(3, 5) are not a function because 5 shows up twice.

    The rule was remembered as "no repeats" instead of "no input with two outputs", so any repeated number looks like a mistake.

    What helps. Ask, "If I put in 2, do I know what comes out?" Then ask the same about 3. If every input has one answer, it is a function.

  • Asked for the output when the input is 1, your child finds 1 on the vertical axis of the graph and reads across.

    The child has not yet tied inputs to the horizontal axis and outputs to the vertical axis, so either axis looks like a fine place to start.

    What helps. Have your child put a finger on the input along the bottom, slide up to the graph, then slide across to read the output. Say "in along the bottom, out up the side" each time.

  • Your child says a table cannot be a function because there is no equation.

    In earlier grades a "rule" usually meant a formula like "add 3", so a list of pairs with no pattern does not look like a rule at all.

    What helps. Make a list of each person in your family and their birthday. Each person has exactly one birthday, so the list is a function even though no formula makes it.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    What is the output when the input is 1?
    123451234560xy
    A
    1
    B
    2
    C
    3
    D
    5
    Show the answer and how to solve it

    The answer is 2 because the point on the graph with x = 1 has y = 2.

    1. 1Find 1 on the x-axis. Then move straight up until you reach the graph.
    2. 2The graph meets that vertical line at the point (1, 2).
    3. 3So the output when the input is 1 is 2.

    Common mistake. A common mistake is choosing 1 because it is the input x-value. After you find x = 1 on the graph, you still need to read the y-value at that point.

    The idea underneath. To find an output from a graph, first match the x-coordinate; the y-coordinate of that same point is the output.

  2. 2On level
    Look at the ordered pairs below. When the input is 5, what is the output?
    {(3,7),(0,2),(5,−3),(1,0),(−2,4)}\displaystyle \{(3, 7), (0, 2), (5, -3), (1, 0), (-2, 4)\}
    A
    5
    B
    3
    C
    2
    D
    -3
    Show the answer and how to solve it

    The answer is -3 because the ordered pair with input 5 is (5, -3).

    1. 1Find the ordered pair whose first number is 5.
    2. 2The matching pair is (5, -3).
    3. 3In an ordered pair, the second number is the output. So the output is -3.

    Common mistake. A common mistake is using the pair (0, 2) and choosing 2. That pair is near (5, -3) in the list, but the question asks for the output when the input is 5, so you need the pair that starts with 5.

    The idea underneath. In an ordered pair, the first coordinate tells you which input to match, and the second coordinate gives the output.

  3. 3On level
    Look at the function table and the query below. Type the matching output.
    xy
    -53
    -1-7
    08
    4-2
    105
    y when x=−5\displaystyle y \text{ when } x = -5
    Output:
    Show the answer and how to solve it

    Answer. 3

    The answer is 3 because the row with x = -5 has y = 3.

    1. 1The query asks for the y-value when x = -5.
    2. 2Look down the x-column in the table and find -5.
    3. 3Read across that row. The matching y-value is 3.
    4. 4Type 3 as the output.

    Common mistake. A common mistake is typing -5 because it is the input named in the query. First find x = -5, then read the y-value from that same row.

    The idea underneath. In a function table, each row is one ordered pair, so once you find the input, the output is the value paired with it in the same row.

  4. 4Stretch
    Sam says, "It is a function because the outputs are all different." Jordan says, "It is not a function because 3 maps to both 7 and -1." Who is correct?
    3→75→23→−16→4\displaystyle \begin{array}{rcl}3 &\to& 7\\5 &\to& 2\\3 &\to& {-1}\\6 &\to& 4\end{array}
    A
    Sam
    B
    Jordan
    Show the answer and how to solve it

    Jordan is correct because input 3 appears in rows 1 and 3 mapping to 7 and -1, which means one input has two different outputs and the relation is not a function.

    1. 1Start on the input side of the mapping and check whether any input value appears more than once.
    2. 2Listing all four pairs, the input 3 appears in row 1 and again in row 3. The two rows are separated by the 5 → 2 row, so the repetition is easy to miss.
    3. 3Input 3 maps to 7 in row 1 and to -1 in row 3. One input pointing to two different outputs means the relation is not a function.
    4. 4Sam's claim is false: all outputs being different does not prevent an input from branching. Jordan correctly identifies the repeated input and its two outputs. Jordan is correct.

    Common mistake. A common mistake is checking whether the output values are distinct and concluding the relation must be a function. The rule is about inputs: if any single input maps to 2 different outputs, the relation is not a function, even when all outputs are different numbers.

    The idea underneath. To test a mapping for function status, scan the input side for one input branching to 2 outputs; repeated outputs are fine.