1. 1. Defining and Representing Functions
Math 1304 Calculus I
1. Defining and Representing Functions

2. Sections Covered 1.1: Four Ways to Represent a Function
1.2: Mathematical Models and Essential Functions
1.3: New Functions from Old

3. Four Ways to Represent a Function Covers functions:
Section 1.1
Four Ways to Represent a Function
Covers functions:
Definition
Terminology
Conceptualization
Ways to represent

4. Definition of Function (of Sets)
Definition: A function is a rule that assigns to each element in one set exactly one element in another set.
f(t) t

5. Terminology Domain – set of values for which the rule is defined
Range – set of values that the rule produces as output
Argument: input to the rule
Value of: output from the rule
Variables
independent variable: input to the rule
dependent variable: output from the rule

6. Conceptualization: arrow diagram
f(x) x
f(a) a A B

7. Ways to represent functions
Verbally – use a language
Numerically – use a table
Visually – use a diagram
Algebraically – use a formula
Implicit: as formula that gives a relation between argument and value
Explicit: value is given directly by a formula in terms of the argument

8. Examples
See book for plenty of examples

9. Real Functions
Note: In this case we study real-valued functions of a real variable.
In other courses we study functions between other types of sets.
Calculus III, functions can go from subsets of n-dimensional space to subsets of m-dimensional space.
In Modern Algebra, functions often go between arbitrary finite sets. Sometimes they go between sets of whole numbers.

10. Graph of a Function
The graph of a real-valued function of a real variable is a curve in the real plane.

11. Vertical Line Test
Vertical line test – a curve in the xy-plane is the graph of a function if and only if no vertical line intersects the curve more than once.
Not a function
Is a function

12. Concepts Symmetry - odd or even functions –
even functions satisfy: f(-x) = f(x)
and odd functions satisfy: f(-x) = -f(x)
Order - increasing/decreasing functions preserve or reverse order.
Increasing: x < y f(x)<f(y)
Decreasing: x < y f(x)>f(y)