1. 1.1 Four ways to represent Functions

2. Definition of a Function

6. Theorem: Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

7. x y Not a function.

8. x y Function.

10. (a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain. (b) f is the symbol that we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range. (c) If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x. Summary Important Facts About Functions

11. Representing Functions

12. Representations of Functions There are four possible ways to represent a function:  verbally (by a description in words)  numerically (by a table of values)  visually (by a graph)  algebraically (by an explicit formula)

13. Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Verbally (with words) or With Diagrams:

14. Numerically: using Tables -

15. Visually: using Graphs -

17. Algebraically: using Formulas – There are several Categories of Functions: 1)Polynomial functions (nth degree, coefficient, up to n zeros or roots) 2)Rational Functions: P(x)/Q(x) – Define domain. 3)Algebraic functions: contain also roots. Ex: f(x)=Sqrt(2x^3-2) or f(x)=x^2/3(x^3+1) 4)Exponential functions: f(x)=b^x ; b: base, positive, real. 5)Logarithmic functions: related to exponentials (inverse), log b x – b: base, positive and not 1. Most common: Exponential base e (2.718…) and inverse: Natural Log. 6)Trig. Functions and their inverses.

19. Piecewise-defined Functions: Example:

20. The absolute value function is a piecewise defined function. Recall that the absolute value of a number a, denoted by | a |, is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have | a |  0 for every number a For example, | 3 | = 3 | – 3 | = 3 | 0 | = 0 | – 1 | = 1 | 3 –  | =  – 3 Important reminders about Absolute Value:

21. In general, we have (Remember that if a is negative, then –a is positive.) Absolute value function f (x) = |x| x if x  0 |x| = –x if x < 0

22. Symmetry: Even and Odd Functions A function f is even if for every number x in its domain the number -x is also in its domain and f(-x) = f(x) A function f is odd if for every number x in its domain the number -x is also in its domain and f(-x) = - f(x)

25. The inverse of f, denoted by f -1, is a function such that f -1 (f( x )) = x for every x in the domain of f and f(f -1 (x))=x for every x in the domain of f -1 : Inverse Functions Theorem The graph of a function f and the graph of its inverse f -1 are symmetric with respect to the line y = x.

26. Increasing and Decreasing Functions The graph shown below rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval [a, b], decreasing on [b, c], and increasing again on [c, d]. Figure 22

27. Example In this graph the function f (x) = x 2 is decreasing on the interval (–, 0) and increasing on the interval (0, ). Figure 23