Functions: A function is a special kind of relations where any "input" gives us exactly one "output."
Example: Consider the function $f(x)=\frac{1}{3}x-2$.
If we hand our function the value $3$, $f$ hands us back $-1$ since $f(3)=\frac{1}{3}\cdot 3 -2=1-2=-1$.
This gives us an ordered pair $(3,-1)$.
We can do this for many different values of $x$:
$$ \begin{array}{c|c} x & f(x) \\ \hline -6 & \\ \hline -3 & \\ \hline 0 & \\ \hline 3 & -1 \\ \hline 6 & \\ \hline \end{array} $$
Functions as Models Functions model many real world phenomena.
Example: the speed of a free-falling object is a function of time.
Example: the price of a stock is a function of time.
Example: the force of gravity is a function of distance.
Functions as Models
Example: A board 18 ft long has three pieces each $x$ ft long cut off. Write a function for the length of the remaining piece in terms of $x$.
Functions as Models
Example: A restaurant automatically adds an 18% gratuity to the food and beverage total on all bills.
a. Write a function for the gratuity added to a food and beverage total of $x$ dollars.
b. Evaluate and interpret $f(40)$.