# Common Core Standards: Math

### Functions 8.F.A.1

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.)

In our super-cyber information age, virtually everyone can work a computer. We have preteens learning code, preschoolers using iPads, and even grandma has branched out from solitaire to Facebook and Twitter ("@PriceIsRight just isn't the same without Bob Barker #disappointment").

In fact, many of your students are probably familiar with the computer science acronym GIGO. It stands for, "Garbage In, Garbage Out," meaning that a computer will do exactly what you tell it to do—not what you want it to do. If you put garbage in, you'll get garbage out.

In math, there are equations called functions that act a lot like computers. A function is a mathematical rule that assigns exactly one y value (the "output") to each x value (the "input"). The thing to remember about functions is that each x can have one and only one y. Though, interestingly enough, each y can have lots of x values, if you want.

Students should understand the concept of functions as having one output for every input. They should also be able to find the output value for a particular x value by plugging the number into the function wherever it has an x. It's also important that they know that the graph of a function represents all the ordered pairs (the different inputs-output combinations) that satisfy the function rule.

Warn your students not to try inputs and outputs of garbage. Those won't satisfy anything.

#### Drills

1. The graph of a function is a set of what?

Points

Since each x value has a corresponding y value, the ordered pairs can be graphed as points. Not all functions are straight lines and circles certainly wouldn't work for functions. It's true that numbers are involved (after all, this is math we're talking about), graphing numbers on the coordinate plane could prove a little difficult. We graph points, which are ordered pairs of numbers.

2. Which of the following does not assign only one y value to each x?

x = 7

This one might be a bit tricky. While (A) and (B) seem like perfectly reasonable functions, we might be wary of (C) and (D). Even though it looks like something we haven't seen before, (D) is a function because we only get one y for every x. Option (C), on the other hand, has no y at all. That means x always equals 7, no matter the y value. So when x is 7, y can be 7 or 2 or 0 or -301. Lots of y values for the same x is not a function.

3. Which set of points does not form a function?

(1, 1), (1, 2), (1, 3)

In order to be a function, the same x value cannot have multiple y values. The points in (A) and (B) don't repeat x or y values, so they can be considered functions. Even (C) repeats only y values, suggesting the function y = 12. That's okay, though, because for every input x, we'll still get one output (y = 12). Only (D) can't be a function because x = 1 produces three different outputs.

4. The equation of a function is y = 6 + x. What is the output when the input is 2?

8

Remember that for functions, the input is always x and the output is always y. So plugging in 2 for x gives us y = 6 + 2 = 8. So our output is 8. The end.

5. The equation of a function is y = 5x – 1. What is the output when the input is 10?

49

Plug 10 in for x, and we get y = 5(10) – 1 = 50 – 1 = 49. So (C) is our answer. All other answers result from inputting the wrong value such as (A), where x = 1 leads to y = 5, subtracting 10 instead of 1, or incorrectly interpreting the equation as 5(x – 1) rather than 5x – 1.

6. The equation of a function is y = x2 – 3. What is the output when the input is 0?

-3

We input 0 for x and find the output y using the given equation. Hopefully it's obvious that 0 squared is still 0. Subtracting 3 from 0 gives us -3. So y = 02 – 3 = -3 is our output. All other answers are incorrectly calculated.

7. The equation of a function is y = x + 5. If the output is 11, what was the input?

6

We know that the input is always x and the output is always y. So if our equation is x + 5 = y and our output is 11, then x + 5 = 11. All we need to do in order to solve for x is subtract 5 from both sides and end up with x = 6. The other answers result from improper calculation.

8. The equation of a function is y = 2x + 1. If the output is -7, what was the input?

-4

We can substitute y = -7 into the equation and then solve for x. If we have 2x + 1 = -7 and we subtract 1 and then divide both sides by 2, we should end up with x = -4 as our answer. Incorrectly substituting -7 for x instead of y or adding 1 rather than subtracting it would give the other wrong answers.

9. Which of the following shapes could not be the graph of a function?

A vertical line

We know that ordered pairs are points, so we can automatically take that off our list. We know that lines can be functions, but are there certain lines that can't be? Remember that the x-axis is horizontal and the y-axis is vertical. If a line is vertical, the same x has a lot of different y values. That means it's not a function. You can also remember that horizontal line has the equation y = c (which is a function) and a vertical line takes the form x = c.

10. The equation of a function is y = x2. If the input is -4, what is the output?

16