Functions and graphs > Functions

With a formula such as $y=\mathrm{-}{x}^{3}+4x$ 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 $y\left(x\right)=\mathrm{-}{x}^{3}+4x$.

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.

$y\left(1\right)=\mathrm{-}{1}^{3}+4\cdot 1=3$ 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:

$y\left(x\right)=\mathrm{-}{x}^{3}+4x=0$.

In this case these values are
$x=0$,
$x=2$ and
$x=-2$.

The corresponding x-intersects are
$(0,0)$,
$(-2,0)$ and
$(2,0)$.