Function: Lecture 1

By the end of the lecture, you are supposed to know the followings:

1. What is a function?

• Can you give a definition?
• Given a graph or an expression, do you know how to check whether it is a function?
• How do you know whether two functions are identical?

2. What are the domain and range of a function?Given a function, how can you find out the domain and the range?
3. What is a restriction of a function?
4. What is maximal domain and how to find out the maximal domain given the function?

Some learning points in this lecture

Notations:

[a, b] This is usually called a closed interval since both the end points a and b are included in the range.

(a, b) This is usually called an open interval since both the end points a and b are NOT included in the range.

[a, b) and (a, b] These two are called half-open-half-closed intervals.

Remark: If one end of the interval goes to infinity (either positive or negative), that end is always denoted with the open bracket ")" or "("

\[D_{f}\] denotes the domain of function f

\[R_{f}\] denotes the range of function f


Concepts

1. A function maps one element in X to one and ONLY one element in Y. But two different elements in X are allowed to be mapped to the same element in Y.
(Think about this point carefully! Can you give an example of the functions that map two different values of X to the same value of Y?)

2. Based on point 1, we can have vertical line test to check whether a graph is a function.

• If any vertical line cut the graph at only one point, it is a function. (one value of X to only one value of Y)
• If there exists a vertical line cutting the graph at more than one point, it is not a function. (one value of X to many values of Y)

3. Two functions are the same if and only if they have both the same rule and domain.

4. If the domain of a function is not specified, it is taken to be the maximal domain for which the function is defined, i.e. x takes whatever values that it is allowed to take.

G.C Skills

How to graph a function with the domain?

Remark: You have to take note of the END POINT in the graph to see whether it is included. The range can be read from the graph.

Tutorial question to attepmt

You can attempt Tutorial Q1 after today's lecture.