1. 2-1 Functions We’ll put the Fun back into Functions!!

2. What is a Function? Definition: The set of (x, y) pairs such that each x has its own unique y value. The x values of a function are called the Domain of the Function and all the y values are called the Range of the Function. X is called the “independent variable” while y is the “dependent variable”

3. Uh, what? That whole each x has its own unique y value can be confusing. So instead, lets use the Birthday Example to explain functions. Get it? Anyone know the vertical line test?

4. One to One Function A one to one function is a function that has one more characteristic; not only does each x have its own y value, no x’s share the same y value. So, is the Birthday Example also a one to one function? What kind of “line test” would prove a one to one function?

5. Well, what would a non function look like? is not a function. Why? So, would indicate a non-function? Anything that is not a function would allow for multiple y values. 99% of the time it would be y^even power OR a ±x.

6. Function Notation The algebraic expression is a function. There are LOTS of functions out there (any equation you can dream up where an x will produce only one y value is a function) but I am going to use this one for now. To show that something IS a function, it is written like this: Don’t worry! The y is replaced by f(x) because y and f(x) are the very same thing!!

7. Remember That!! y  f(x) f(x) simply tells the reader that the relation given is more than simply a way of getting x & y pairs. These x & y pairs fit the definition of a function!! You also might see f(x), g(x), h(x) etc. In physics, there are functions listed as s(t), v(t), and a(t). In Calculus typical function notation might be V(t), h(t), or A(x). The variable inside is the independent variable, and the variable outside represents the “y” (or range) value.

8. OK – how do we use it? Lets use the sample from before. 1.Given find f(1), f(-2) and f(0). The function is simply an instruction of what to do to x. Lets find the answers and write them as x/y pairs for now.

9. Examples 2.Given find f(5), f(12) and f(0). This last problem leads us to the “domain” problem. Specifically, you will be also asked to find the domain of a function.

10. Domain? What was that? Oh yeah – the x values. The easiest way to define the domain (all the x values possible) is to define what the domain can’t be. For example 2, what is a disallowed value for the function? If x ≠ 0, then typically you say either the domain is a)All real numbers except 0 OR b) x ≠ 0 Either is acceptable.

11. Examples Find the domain of 3. 4. 5.