A function works such that you end up with a relationship between variables. It works just like a machine. You put something in, and something new comes out.

Imagine a bread maker machine. If you put in finely ground, bleached flour, you end up with a smooth, white loaf. If you put in seeded, rough flour, you get multigrain wheat bread. The difference between a function and a bread maker is that you put numbers into a function—not flour!

Two examples of functions are $y=x+2$ and $f(x)={x}^{2}-2$. The function $y$ is the straight line in Figure (b), and $f(x)$ is the graph in Figure (d) below.

As you can see from the two functions in this section, one begins with $y$ and the other with $f(x)$ (read as “$f$ of $x$”). Why? Functions can have different names, some of the most common ones are $y$ and $f(x)$. Both tell you that they are values from the second axis.

The notation $f(x)$ tells you that you have a function that depends on the $x$-value. That is, you put the numbers from the $x$-axis into the function, and the value you get is a number on the $y$-axis. Since $f(x)$ gives you $y$-values, you can think of them as equals, so that $y=f(x)$. A function thus receives a value, and returns a value.

You can insert many $x$-values into a function, while the $y$-values are directly dependent on what the $x$-values are. We therefore call the $x$-value the independent variable and the $y$-value the dependent variable.

Theory

For each value of $x$ there is only one value of $y$.

Above you see four figures. Figures (b) and (d) show graphs that are functions. From the figures you can see that each $x$-value has only one corresponding $y$-value.

In Figures (a) and (c) you see that an $x$-value can have several different $y$-values. These are thus not functions.

In Figure (a), the $x$-value we chose has two $y$-values, which is true of almost any x value in a circle. In Figure (c) the $x$-value we chose has four $y$-values. We call these types of figures curves.

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