Function rule

Functions: Domain and range

Function rule

We have just seen that a function can have a corresponding formula. From now on, we will also give functions a name. This can be convenient if we are dealing with multiple functions. It helps us easily identify which function we mean.

We can notate a function with the help of a function rule. We then name the function, for example, #\orange f#.

A function with formula #y=4 \cdot \blue x +2# has the following function rule:

\[\orange f(\blue x)=4 \cdot \blue x +2\]

Since, if #\blue x=\blue2# then #\orange f(\blue2)=4 \cdot \blue2 +2=\green{10}#, we can say that the #\green{\text{function value}}# of #\blue 2# is equal to #\green {10}#.

LettersIn the definition, we used #f# as the function's name. We can use every letter possible. In general, #f#, #g#, or #h# are most often used.

With a function rule we can also notate functions that do not have a formula of the form #y=\ldots#. Take a look at the example.

Example

\[f(x)=\left\{\begin{array}{ll}0 & \text{for } x\lt0 \\ 1 & \text{for } x \geq 0\end{array}\right.\]

Take a look at the function:
\[f(x)=-9\cdot x^3+8\cdot x^2-7\cdot x-7\]
Calculate #f(-3)#.

#f(-3)=# #329#

After all, to calculate #f(-3)#, we substitute #x=-3# in the function.
We then get: \[f(-3)=\left(-9\right)\cdot \left(-3\right)^3+8\cdot \left(-3\right)^2+ \left(-7\right)\cdot \left(-3\right)-7=329\]
Hence, #f(-3)=329#.

New example