1. AP Calculus Chapter 1 Section 2
Functions and Graphs

2. Functions
The values of one variable often depend on the values of another: i.e. the area of a circle depends on the circle’s radius.
The area A is dependent on the variable r. From this we can assign the names of dependent variable (A) and independent variable (r).

3. Functions
A rule that assigns to each element in one set a unique element in another set is called a function.
The sets may be sets of any kind and do not have to be the same.
A function is like a machine that assigns a unique output (y) to every allowable input (x).
The inputs make up the domain and the outputs make up the range.

4. Functions
Definition: A function from a set D onto a set R is a rule that assigns a unique element in R to each element in D.
Euler invented the symbolic way to say
“y is a function of x”: y = f(x); which is read “y equals f of x”.

5. Domains and Ranges
The domain of the function is restricted by context: the independent variable is a radius and must be positive.
When we define a function y = f(x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y-values: this is called the natural domain.

6. Domains and Ranges If we want to restrict the domain, we must say so.
The domain of “y = x2” is understood to be the entire set of real numbers.
We must write “y = x2, x > 0” if we want to restrict the function to positive values of x.

7. Domains and Ranges
The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals.
The intervals may be open, closed, or half-open and finite or infinite. See figures 1.10, 1.11, & 1.12 on page 13.

8. Domains and Ranges
The endpoints of an interval make up the interval’s boundary and are called boundary points.
The remaining points make up the interval’s interior and are called interior points.
Closed intervals contain their boundary points.
Open intervals contain no boundary points.
Half-Open intervals contain one boundary point.

9. Domains and Ranges Identifying Domain and Range from a Graph
See Example 2 on page 14 and example 3 on page 15
Graph Viewing Skills
Recognize that the graph is reasonable
You need to know the basic functions, their graphs, and how changes in their functions affect their graphs.
See all the important characteristics of the graph
Interpret those characteristics
Recognize grapher failure
Occurs when the graph produced is less than precise due to limitations of the grapher.

10. Even & Odd Functions
The graphs of even and odd functions have important symmetry properties.
Definition: A function y = f(x) is an
Even function of x if f( -x) = f(x)
Odd function of x if f( -x) = -f(x)
For every x in the function’s domain

11. Even & Odd Functions The names even & odd come from the powers of x.
The graph of an even function is symmetric about the y-axis.
Since f( -x) = f(x), a point (x, y) lies on the graph if and only if the point ( -x, y) lies on the graph. See figure 1.15(a) page 15
The graph of an odd function is symmetric about the origin.
Since f( -x) = -f(x), a point (x, y) lies on the graph if and only if the point ( -x, -y) lies on the graph. See figure 1.15(b) page 15

12. Piecewise Functions - Functions Defined in Pieces
While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domains.
Example:

13. Absolute Value Function
The absolute value function y = |x| is defined piecewise by the formula

14. Composite Functions
Suppose that some of the outputs of a function g can be used as inputs of a function f.
We can then link g and f to form a new function whose inputs x are inputs of g and whose outputs are the numbers f(g(x)).
We say that the function f(g(x)), read f of g of x, is the composite of g and f.
The stand-alone notation is f ∘ g

15. Composite Functions
Work on Exploration 1 on page 18 in your book.

16. Homework
Exercises #3 – 48 by multiples of 3, and #54, 55, 56.