Imagine a machine, like the one above, where you put something in one end, and something comes out the other end. If you put a hunk of metal on a conveyer belt, and don't do anything to it, the same hunk of metal will come out the other side unchanged. The same thing will happen to a number if you put it into the input-output machine and multiply it by one -- or add zero to it. Take the number five for example. Put five into the left side, multiply it by one, and five comes out the right side (because 5 x 1 = 5). Or put a six into the machine and add zero to it, and a six will come out the right side (because 6 + 0 = 6). Multiplying a number by one or adding zero to a number does not change that number, and this is known as the Identity Property. To change that hunk of metal, we could heat it up, shape it into something else, then cool it off. To change a number inside the input-output machine, we use
a function rule. For example, we may use the rule "3x + 2" which states that we will input a number, multiply it by three, and add two to it. Let's try a seven. Input seven into the left side of the machine and apply the rule: 3(7) + 2 = 23, so the output 23 comes out the right side of the machine.
The variable x, which is loaded into the left side, is called the independent variable.
It is independent because it does not depend on anything else. You could input a 7, like we did above, or an 8, 9 or 99. For that matter, you can input a negative number, a fraction or a decimal. The output, on the other hand, is called the dependent variable, because it depends on what number is input and how the function rule affects it. Using the same function rule that turned a 7 into a 23, we could input an
8 and get a 26, or a 10 and get a 32. For each number that we input, there is exactly one number that can be output.
A function table can be used to help calculate how the numbers that are input are turned into the numbers that are output. It can even help calculate backwards to
use the output to find the original input. Here is an example of two function tables, one that employs the function rule "x + 6" in a horizontal format, and one that shows the rule "2x + 1" in a vertical format.
a function rule. For example, we may use the rule "3x + 2" which states that we will input a number, multiply it by three, and add two to it. Let's try a seven. Input seven into the left side of the machine and apply the rule: 3(7) + 2 = 23, so the output 23 comes out the right side of the machine.
The variable x, which is loaded into the left side, is called the independent variable.
It is independent because it does not depend on anything else. You could input a 7, like we did above, or an 8, 9 or 99. For that matter, you can input a negative number, a fraction or a decimal. The output, on the other hand, is called the dependent variable, because it depends on what number is input and how the function rule affects it. Using the same function rule that turned a 7 into a 23, we could input an
8 and get a 26, or a 10 and get a 32. For each number that we input, there is exactly one number that can be output.
A function table can be used to help calculate how the numbers that are input are turned into the numbers that are output. It can even help calculate backwards to
use the output to find the original input. Here is an example of two function tables, one that employs the function rule "x + 6" in a horizontal format, and one that shows the rule "2x + 1" in a vertical format.