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