For at least one of the tables, describe a second rule that corresponds to the given pairs, but ultimately produces different pairs from the first rule for the rest of the table. The task is reminiscent of a function that has more value as a puzzle than as a mathematical value. It takes positive integers as input and has as output the number of letters in its English (American) notation, so that the first values are $3,3,5,4,4,3,5,ldots$. We have a mobile plan that asks when you sign up and takes per month. What is the cost function? Draw multiples of any number on a numeric line. We will use multiples of 3. So we plot the values of 3x on the numeric line. Since we know that multiplication is just repeated addition, the common difference between multiples is 3. Now consider adding or subtracting any of these 3x values. For example, we could add 2 to all multiples of 3. Then the resulting numbers all follow the 3x+2 rule. It is important to note that all numbers move around the same amount, which means that the difference between them does not change. This task can be an opportunity to discuss mathematical modelling and functional adjustment (to give a real example, one can discuss sea level forecasting) as well as the nature of scientific extrapolation and inductive reasoning versus mathematical deductive reasoning.

In this way, we see that the output value is always equal to 4x−1, so the function rule is y=4x−1. We can now use this rule to find the missing values by inserting the values of x into the function. To express the relation in this form, we must be able to write the relation, where [latex]p[/latex] is a function of [latex]n[/latex], which means that it is written as [latex]p=[/latex] expression with [latex]n[/latex]. If we have a function in formula form, it is usually simple to evaluate the function. For example, the function [latex]fleft(xright)=5 – 3{x}^{2}[/latex] can be evaluated by the square of the input value, multiplying it by 3, and then subtracting the product from 5. Fill in the input-output table for the function y=5x+3. Since this is clearly not the only operation that fulfills the function, we know that the rule must have at least two steps. We can add an intermediate row to our input-output table to calculate the result of multiplying the input by 2, and then compare those values to the output to see what other calculations are done. When we do this, we see that we only have to add 1 to go from 2x to the output. In the following video, we will give another example of how to solve a function value. For examples like the first part, a question like “Could we define a function with different letters in the word?” might arise.

It is clear that “take the first letter” is a rule that defines a function from words to letters. But to do something like “take the third letter” when some words don`t have a third letter, we have to worry about accuracy. There are a number of reasonable options. Either the sentence from which the entry is taken can be modified to contain words of at least three letters. Or we can add an element to the output set to indicate the absence of a letter (mathematicians would normally use $phi$ to specify such a “zero letter”). You can also change the rule so that, for example, the last letter is used for words with less than three letters. There is no mathematical reason to prefer one or the other of these elements, but modeling situations often point to one or the other. For example, in Scrabble, each word has at least two letters, so a function for the second letter would be well defined. We can see that the letters provided as output are the last letters in the words provided as input, so a possible rule is “the last letter” of the input.

Here is a way to complete the table according to this rule. Write a function rule for “Output is half of input.” The domain of the function is the type of pet and the range is a real number that represents the number of hours that the duration of storage of the animal lasts. We can evaluate the [latex]P[/latex] function at the input value of “goldfish”. We would write [latex]Pleft(text{goldfish}right)=2160[/latex]. Note that to evaluate the function in tabular form, we identify the input value and the corresponding output value from the corresponding row of the table. The tabular form of the [latex]P[/latex] function seems to be ideal for this function, more so than writing as a paragraph or function. For example, the function y=x+1 has the variable x as input and the variable y as output. Solution: This means that we evaluate the function if x is set to 2. The first step is to replace each x with 2.

Then evaluate the function according to the order of operations (BEDMAS). Therefore, our rule is 2x+1, which is y=2x+1 as a function that connects x and y. Note that you have to add 4 to go from 6x to output y. Therefore, we conclude that the function rule y = 6x + 4. It is still true that if the common difference between successive outputs is a constant, the function begins by multiplying the input by a.