AP CALCULUS Review-2: Functions and their Graphs

FunctionsDef: A function is a special type of relation.

y is a function of x if for each x-value there is only one y-value. The graph of a function passes the vertical line test. This is written

• Domain: the set of all x-values • Range: the set of all y-values

Notation:

)(xfy

Functions

One-to-one Function: a function in which not only is there only one y for each x, but there is also only one x for each y. The graph passes the horizontal line test as well as the vertical line test.

Find the domain and range

1.

f (x) x 4

Find the domain and range

2.

g(x) sin x

Functions are commonly represented 4 ways

Verbally: by a sentence that describes the relationship between variables

Numerically: by a table or list of ordered pairs

Graphically: by points on a coordinate plane

Analytically: by an equation in two variables

Find x-intercepts and the y-intercepts

3.

f (x) x 3 3x 2 2x 5

Piecewise Functions

 Piecewise Function: a function defined differently on different pieces of its domain.

 

 

Example:

Piecewise Functions

4. Find the domain and range for

01

01)(

2

xx

xxxf

Parent Graphs

a) Constant function

b) Cubic Function

Graph

5.

what is the transformation?

12)(

)(3

3

xxf

xxg

Behavior of a graph

If a graph is continuous it has No breaks, holes or asymptotes

Leading coefficient Test

In the polynomial function

Where the a, b , and c values are coefficients and a is the leading coefficient

If a is not zero then n is the degree of the polynomial

f (x) ax n bx n 1 cx n 2...

End Behavior is determined by the degree

a. Even degree

b. Odd degree

c. Notation for end behavior

6. Is the polynomial even or odd?

7. Graph and Describe the end behavior of the graph

f (x) x 3 4x

8) Graph and describe the end behavior

y x 4 5x 2 4

9. Find the discontinuity, then graph

2

42

x

xy

Additional exercises

Review Assignment 2

Problems # 10-14