# Functions that Describe Situations

## Key Questions

• Suppose we were asked to find the volume of a balloon

we will use the function of volume for a balloon

$v \left(b a l l o n\right) =$4/3 pi r^3

$\frac{4}{3} \cdot \pi {3}^{3}$

$4 \pi \cdot 9$

$36 \pi$

Hence there several functions which are much more complex than this.

• Consider a taxi and the fare you have to pay to go from A street to B avenue and call it $f$.

$f$ will depend upon various things but to make our life easier let assume that depends only upon the distance $d$ (in km).
So yo can write that "fare depends upon distance" or in mathlanguage: $f \left(d\right)$.

A strange thing is that when you sit in the taxy the meter already shows a certain amount to pay...this is à fixed amount you have to pay no matter the distance, let's say, 2$. Now for each km travelled the taxi driver has to pay petrol, maintenance of the vehicle, taxes and get money for himself...so he will charge 1.5$ for each km.
The meter of the taxi will now use the following function to evaluate the fare:
$f \left(d\right) = 1.5 d + 2$
This is called "linear" function and allows you to "predict" your fare for each distance travelled (even if $d = 0$, i.e., when you only sit in the taxi!)
Now, let us assume that the distance $d$ between A street and B avenue is $d = 10 k m$, you fare wiĺ be:
f(10)=1.5×10+2=17\$

You can now improve your function including additional costs and dependences or build new relationships.