Functions relate dependent and independent variables, where the dependent variable is determined by the independent variable. Functions can be expressed as equations, tables, or graphs. Even functions are symmetric about the y-axis and odd functions are symmetric about the origin. The domain of a function is the set of valid inputs and the range is the set of outputs. A function can be restricted to only certain domain values or composed of multiple functions.

1. Functions and Graphs Unit 1 Class 2 E. Alexander Burt Potomac School

2. Functions Simplest definition: the value of one variable depends on another variable The dependent variable is the one that is determined by...

3. The independent variable The domain of the function is the set of permitted values for the independent variable

4. The range of the function is the set of permitted values for the dependent variable.

5. Set definition of a function A function is a rule which maps one set (the domain) to another set (the range) One domain value can only produce one range value

6. The same range value can be produced by several domain values. Example: y = x 2 X is the independent variable

7. X = + or – 2 will yield y=4

8. Ways of expressing functions Equations: this is probably the most common

9. f(x) = p x 2

10. Tables

11. Graphs In general the horizontal (abcissa) axis is the independent variable (and thus, the range)

12. And the vertical (ordinate) axis is the dependent variable (and thus the domain)

13. Even and Odd Functions A function is even if f(-x) = f(x) Even functions are symmetric about the y axis

14. If even functions are polynomials, the largest power will be an even number (get it?) A function is odd if f(-x) = f(x) Odd functions are symmetric about the origin

15. An odd power polynomial (x 3 for example) is an odd function. Polynomials are not the only even and odd functions!

16. Restricting Domain and Piecewise Functions It is possible to restrict the domain of a function by stating the valid domain (show examples in equations and graphs)

17. Putting together several functions with restricted domains, you get a piecewise function. Ex: absolute value function

18. Composite functions If the independent variable in a function comes from another function you have a composite function:

19. f(g(x))

20. Evaluate the innermost function first