Functions and graphs > Functions
12345Functions

## Theory

With a formula such as $y=-{x}^{3}+4x$ you will find exactly one value for $y$ for every value of $x$. In that case, $y$ is a function of $x$ with function rule $y\left(x\right)=-{x}^{3}+4x$.

When you are given a function, you can compute a table and draw the corresponding graph. The values of the input variable are found on the horizontal axis, the $x$ -axis.
The results are called function values.
$y\left(1\right)=-{1}^{3}+4\cdot 1=3$ is the function value if $x=1$. Function values are found on the x -axis.

You also find the expression $f\left(x\right)$ being used for $y\left(x\right)$. $y$ then is a function of $x$ that is given the name $f$.
In contextual situations you often use letters that have some connection to the meaning or name of the variable. For time you use $t$, for length you use $l$, for volume you use $V$, for velocity you use $v$, for power you use $P$, etc. Your graphing calculator is set to use X as the standard input variable, and Y as the standard function value.

The zeros or roots of a function are those input values that give you a function value of $0$. For the function above these are the y -values for which:
$y\left(x\right)=-{x}^{3}+4x=0$.
In this case these values are $x=0$, $x=2$ and $x=-2$.
The corresponding x-intersects are $\left(0,0\right)$, $\left(-2,0\right)$ and $\left(2,0\right)$.