1. Introduction to Functions
Section 8-7: Functions Defined by
Equations

2. The “Rules” of the Game
You learned how to define a function by drawing a chart or graph.
This is helpful because it let’s you see what a function looks like over a small set of data.
However, this can be a bit more difficult for large sets of data.
We can define a function with a ‘rule’ that works for any data set (big or small)
This ‘rule’ is an equation.

3. Our First Example: CBHS presents Treasure Island
Let’s say they charged $5 for each ticket.
Imagine that it cost the school only $500 to put on the production.
The school’s profit depends on how many tickets they sell.
Let p be the profit, and let n be the number of ticket sold.
The equation p = 5n is a function describing the relationship between profit and the number of tickets sold.

4. Domain and Range for our function
Domain: the number of tickets sold
The possibilities are that the school could sell 0 tickets, 1 ticket, 2 tickets, …
So the Domain is D={0,1,2,…}
Range: the profit
If nobody went to the play, their profit would be -$500. (The cost of production)
For each ticket sold, this profit would increase by $5.
So the Range is R={-500, -495, -490,…}

5. You find the Range
Using “Arrow Notation” we have the function F: x x+5. For the domain,
D = {0, 1, 2}, what is the Range?
0+5=5, 1+5=6, and 2+5=7
So R = {5, 6, 7}
Find the range of F: x x+5 for the domain D= {-1, 3, 5}
D = {-1, 3, 5} R = {4, 8, 10}

6. Our Variables
Since our function has two variables, we need to distinguish between their roles.
The profit ‘depends’ on how many tickets are sold, so we call the profit the Dependent Variable.
The number of tickets sold doesn’t depend on any other variable, so we call the number of tickets sold the Independent Variable.

7. Function Notation
We can write out this relationship using function notation, which distinguishes between the dependent and independent variables.
For our example, the equation p = 5n can be written using the function notation
P(n) = 5n – 500
This reads, “P of n equals 5n – 500”.
In this example, the function assigns a value to the number of tickets sold.
The values assigned to each independent variable make up the Range of the function.

8. Function Notation (continued)
Function notation is useful because we can find any solution to this equation by just plugging in a given independent variable. (This means we can find the range of a function by determining the values for each independent variable.)
Ex: f(x)= x2 - 2x
This means f is a function of the variable x.
What is f(4)? What is f(-3)?
f(4) = ·4 = 16 – 8 = 8
(when x is 4, the function has a value of 8.)
f(-3) = (-3)2 - 2·(-3) = 9 – (-6) = = 15
(when x is -3, the function has a value of 15.)

9. Function Notation or Arrow Notation?
Both ways of representing functions do three things:
Give the name of the function
Tell which variable the function uses
Define the function with an equation
Note: Function Notation is used more frequently than Arrow Notation.

10. Review of Linear Functions
We’ve learned about one type of function already: the line.
Remember the slope-intercept form??
y=mx +b
Well, this is a function whose domain is the set of possible x values, and whose range is the set of possible y values.
You can represent a function with a graph as in 8-6, or with an equation like this.
The solutions to this equation are ordered pairs where the independent variable is x and the dependent variable is y.

11. Conclusion
When a function is defined by an equation, the domain of the function must also be given. By substituting a value of the domain in the equation, the corresponding value of the range is found. (if you’re given the x, you can find the y.)
Homework:
Pages #’s 3-24 multiples of 3