# Functions: Describing Behavior

A function table (T-table) shows you the relationship between pairs of numbers. The relationship is defined by a rule, and this rule applies to all the pairs of numbers on the table.

The functions can be linear or nonlinear.

First look for the pattern with the input and output pairs. Then you can use either of the following two methods to start writing the function rule.

- Guess and check.
- What do you have to do with $x$ to get $y$ ?

Try adding or subtracting or multiplying or dividing something to $x$ to get $y$ to write the rule. Once the function rule is ready, make sure the rule works for each set of numbers.

You can also guess the functions behavior from the table of values. When the $x$ -value increase and the $y$ -value increase, then the graph of the function goes up. When the $x$ -value increase and the $y$ -value decrease, then the graph of the function goes down.

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Example:
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Write the function rule from the T-table.

$x$ | $y$ |

$1$ | $4$ |

$2$ | $8$ |

$3$ | $12$ |

$4$ | $16$ |

$5$ | $20$ |

Look at the pair of numbers $1$ and $4$ .

How do you get $4$ from $1$ ? You can add $3$ or multiply $1$ by $4$ .

Now look at the pair of numbers $2$ and $8$ .

How do you get $8$ from $2$ ? You can add $6$ or multiply $2$ by $4$ .

Now look at the pair of numbers $3$ and $12$ .

How do you get $12$ from $3$ ? You can add $9$ or multiply $3$ by $4$ .

Observe the pattern. You can easily get the $y$ -values when you multiply the $x$ -values by $4$ .

So, the function rule is $f\left(x\right)=4x$ or $y=4x$ .