Functions Describing Behavior

We've worked with functions before, but what if you have a list of x and y values and want to know the function rule they correspond to? A function table (alternatively T-table) shows us the relationship between two sets of numbers. That relationship is defined by a rule, and the rule always applies to every single number in the table. Note that the corresponding functions can be linear or nonlinear.

In this article, we'll look at T-tables and discover how to deduce the corresponding function rule. Let's get started!

An introduction to functions describing behavior

The first step in determining a function rule is to look at the inputs and outputs. Once you think you see a pattern, you have two choices:

1. Guess and check

Basically, you're writing the pattern you see mathematically and then plugging each of the values in to make sure it works for every set.

2. What do we have to do with x to get y?

If you aren't sure what the pattern is, look at the x values and ask yourself how to turn them into the corresponding y values. Adding, subtracting, multiplying, or dividing by some constant are the most likely choices. Make sure to double-check your work by plugging each value into the rule you come up with.

You can also guess the behavior of a function by looking at the x and y values. If both values keep increasing, you know the function will go up once graphed. If the y value decreases as the x value increases, you know that the graphed image will go down.

A practical example of functions describing behavior

The best way to study this topic is to try your hand at it, so let's look at a sample problem.

Write a function rule based on the following function table:

x	y
1	4
2	8
3	12
4	16
5	20

Do you see the pattern? If not, you should start by examining the lowest numbers first. How do we get 4 from 1? We could add 3 or multiply by 4. Keep both of those in mind.

Next, how do we get 8 from 2? We could add 6 or multiply by 4.

How do we get 12 from 3? We could add 9 or multiply by 4.

"Multiply by 4" has come up in the first three examples, so it's time to start seriously considering that as our function rule. Multiplying $4\times 4=16$ , so it works for 4 and 16. Likewise, $5\times 4=20$ , so it works for our final set of numbers as well. Now that we know the rule, all we have to do is express it mathematically:

$f\left(x\right)=4x$

We've successfully found our function!

Practice questions on functions describing behavior

a. Find a function rule for the following T-table:

x	y
1	1
2	4
3	9
4	16
10	100

This one's a little trickier. To get 1 from 1, we can multiply by 1, add 0, or subtract 0. However, none of those work for any of the other numbers on the list. Similarly, we can multiply by 2 or add 2 to turn 2 into 4, but neither works for anything else. Worst of all, our last set of values gets into triple digits. Should we assume there is no function rule?

Of course not! While adding, subtracting, multiplying, and dividing are the most common options, anything we can express mathematically is fair game. If we look at 1^2, we're effectively doing $1\times 1=1$ which we already noted satisfies our first set of values. Likewise, $2^2$ is the same as $2\times 2$ and gives us the y value of 4. $3^2=9$ , $4^2=16$ , and even ${10}^2=100$ . We have a workable function rule after all, and we can express it mathematically like this:

$f\left(x\right)=x^2$

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Varsity Tutors can help with functions describing behavior

Figuring out function rules based on a few x and y values requires some guesswork that can feel extremely overwhelming if students aren't sure where to begin. If your student isn't someone who will immediately spot the patterns, a private math tutor can provide step-by-step tutorials and extra practice problems until they gain the self-confidence necessary to succeed. Working in a private learning environment tailored to your student's preferences may also improve their study efficiency. Contact the Educational Directors at Varsity Tutors for more info on what makes tutoring such a powerful educational tool or to sign up today.