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Common Core Standards: Math

High School: Functions

Interpreting Functions HSF-IF.A.1

1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

A function is like any other system. What you get out of the system depends on what you put into it. Think of the human body. We put food into our digestive system and we get something very different out. (Gross.) What we put into our bodies affects what comes out and if you don't believe that, try eating beets or asparagus.

Students should understand that functions do the exact same thing, only with numbers. (Maybe not the exact same thing.) They're all about describing relationships between two sets of numbers. These two sets of numbers must have the condition that each item from the first set of numbers pairs with exactly one item from the second set of numbers.

In other words, for every input, there is exactly one output. That's like the functions' motto.

Students should know that functions can be expressed as a pair of input and output values. A relation is a set of pairs of input and output values, usually represented in ordered pairs. For instance, the ordered pair (1, 2) means that for an input value of 1, we get an output value of 2. The ordered pair (2, 3) means that if we input 2, we get 3 out. Typically, we represent this as (x, y).

 

The domain is the set of inputs in a relation, also called the x-coordinates of an ordered pair. The range is the set of outputs in a relation, also called the y-coordinates of an ordered pair. If students have a hard time remembering which is which, tell them to think alphabetically. Since D comes before R in the alphabet, the domain has to come before the range. If that doesn't work, the acronym "DIXROY" might. (Domain, input, x; Range, output, y.)

 

To start with, students can represent functions as several ordered pairs in the form of a table. One column will be the input, or x values, and the other will be the output, or y values. For instance, we can rewrite the three points of a function (-2, 3), (0, 4), and (1, -3) in the following table.

Here, we have our clearly defined domain (D: x = -2, 0, 1) and range (R: y = 3, 4, -3). Any table with the same x value resulting in multiple y values is not a function. Remember the functions' motto?

When domains and ranges cover more than just a few select points, they're often included in parenthesis or brackets. Parenthesis indicate that the point on that end is not included, while brackets indicate that it is included. When the ∞ symbol is used, we use parentheses. Makes sense, since infinity isn't really a number and can't actually be reached.

As useful as tables are, many functions have domains and ranges that extend to positive and negative infinity. When the students' data tables start getting longer than their arms, we recommend switching to graphs. Spare a headache and a few trees, while they're at it.

A graph is a visual representation of relations. We plot the input values as x and the output values as y and treat the ordered pairs (x, y) as points on the coordinate plane. For the function above, we could plot the points as (-2, 3), (0, 4), and (1, -3).

Students should also know that functions can be represented by curves on the coordinate plane as y = f(x) where f(x) is some function of x. These are basically a bunch of points that are so close together that they form a continuous curve. For instance, these points could be part of a larger function shown by the graph below.

If students aren't sure whether they're looking at a function or not, they should perform the vertical line test: if they draw a vertical line on a graph of a relation and it intersects with the curve more than once, the relation is not a function.

The key concept to remember is that functions are systems in which one input corresponds to one output. Just like the human body is a system in which every meal corresponds to one trip to the bathroom. Or something like that.

 

Drills

  1. Given the ordered pairs (-12, 0), (-6, -1), (2, 3), (8, 5), and (-3, 4), identify the domain.

    Correct Answer:

    {-12, -6, -3, 2, 8}

    Answer Explanation:

    The domain of the function is given by (B) since it lists all x values. Answer (A) lists all x and y-values, (C) lists the range (only the y values), and (D) lists all values that are common to both the domain and range. Only (B) lists the correct values.


  2. Given the ordered pairs (0, 3), (-2, 11), (1, 5), (2, 11), and (-1, 5), identify the range.

    Correct Answer:

    {3, 5, 11}

    Answer Explanation:

    The range of the function is given by (D) since it lists all the y values. Answer (A) lists all the x-values (the domain), and (B) is simply an empty set. Answer (C) lists all the x and y values. Only (D) lists the correct values.


  3. Given the following table of values, identify the range.

    xy
    -23
    04
    1-3
    3-2
    60
    98
    1211

    Correct Answer:

    {-3, -2, 0, 3, 4, 8, 11}

    Answer Explanation:

    The range of the function is given by (A) since it lists all the y values. Answer (B) lists all the values common to both the domain and the range, (C) lists the x-values (the domain), and (D) lists all the x and y values. Only (A) lists the correct values.


  4. Determine which relation is a function.

    Correct Answer:

    xy
    -13
    04
    1-3

    Answer Explanation:

    Since each x value is paired with exactly one y value in a function, the only one that fits the bill is (A). All the other answers have an x value that leads to two different y values. In (B), an input of 4 can create an output of both 3 and 4. This is impossible for a function.


  5. Identify the set of ordered pairs that represents a function.

    Correct Answer:

    (1, -1), (-2, -7), (0, 1), (-1, -1)

    Answer Explanation:

    A function must have one output value for each input value. Since (A), (C), and (D) all contain a value in the domain that pairs with more than one value in the range, they are not functions. Only (B) is a function because each x value corresponds to exactly one y value.


  6. Identify the domain and range of the following graph.

    Correct Answer:

    D: [-2, 2]; R: [0, 4]

    Answer Explanation:

    Look at the graph. The x value (the domain) goes from -2 to 2. The y value (the range) go from 0 to 4. Since they include the points (the endpoints of the line aren't open circles, are they?), we use brackets rather than parenthesis. That makes (A) the right answer.


  7. Identify the domain and range of the following graph.

    Correct Answer:

    D: [-1, +∞); R: [0, +∞)

    Answer Explanation:

    Look at the graph. The x values range from -1 to +∞, meaning the domain is [-1, +∞). The y values start at 1, but they go down to 0 and then up to +∞. That means the range is [0, +∞). The answer we're looking for is (B).


  8. Identify the values that do not represent a function.

    Correct Answer:

    xy
    -10
    00
    1-1
    -1-2

    Answer Explanation:

    The x value of -1 being resulting in y values of both 0 and -2 make the relation in (A) the only non-function. The rest of the answer choices are functions since each value in the domain pairs with exactly one value in the range.


  9. Which point could not be part of a function that includes (-1, 6), (2, 2), (3, 4), (0, -4), and (1, -2)?

    Correct Answer:

    (1, 4)

    Answer Explanation:

    Each value in the domain can only be paired with one and only one value in the range. A point that has an x value of -1, 2, 3, 0, or 1 cannot be part of the function. Answer (D) has an x value of 1, so it cannot be part of the function because a function would not have both (1, -2) and (1, 4) as points. The other answers don't repeat x values, so they may still be part of the function.


  10. Which of the following graphs is a function?

    Correct Answer:

    Answer Explanation:

    We can use the vertical line test to determine which of the graphs are functions and which are not. The only graph that touches a vertical line at only one point is (D), so (D) is the only function.