mat 125 – applied calculus 2.1 functions and their graphs
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MAT 125 – Applied Calculus2.1 Functions and Their Graphs
2.1 Functions and Their Graphs
2 Today’s Class
We will be learning the following concepts: Functions
Determining the Domain of a Function
Graphs of Functions
The Vertical Line Test
Dr. Erickson
2.1 Functions and Their Graphs
3 FunctionsFunction: A function is a rule that assigns to each element in a set A one and only one element in a set B.
The set A is called the domain of the function.
It is customary to denote a function by a letter of the alphabet, such as the letter f.
Dr. Erickson
2.1 Functions and Their Graphs
4 FunctionsThe element in B that f associates with x is written f(x) and is called the value of f at x.
The set of all the possible values of f(x) resulting from all the possible values of x in its domain, is called the range of f(x).
The output f(x) associated with an input x is unique:
Each x must correspond to one and only one value of f(x).
Dr. Erickson
2.1 Functions and Their Graphs
5 Example 1Let the function f be defined by the rule .
. Find 4 . Find 0
. Find . Find
a f b f
c f a d f a h
Dr. Erickson
2
2
1
xf x
x
2.1 Functions and Their Graphs
6 Determining the Domain of a FunctionSuppose we are given the function y = f (x). Then, the variable x is called the independent variable. The variable y, whose value depends on x, is called the dependent variable.
To determine the domain of a function, we need to find what restrictions, if any, are to be placed on the independent variable x.
In many practical problems, the domain of a function is dictated by the nature of the problem.
Dr. Erickson
2.1 Functions and Their Graphs
7 Example 2 Find the domain of the function.
Dr. Erickson
22
2
3 1 3 .
1. 5
4
xf x x f x
x
xf x x f x
x
a. b
c d.
2.1 Functions and Their Graphs
8 Graphs of FunctionsIf f is a function with domain A, then corresponding to each real number x in A, there is precisely one real number f (x).
Thus, a function f with domain A can also be defined as the set of all ordered pairs (x, f (x)) where x belongs to A.
The graph of a function f is the set of all points (x, y) in the xy-plane such that x is the domain of f and y = f (x).
Dr. Erickson
2.1 Functions and Their Graphs
9 Example 3The graph of a function f is shown below:
Domain
Range
Dr. Erickson
a. Find f (0).
b. Find the value of x for which f (x) = 3.
c. Find the domain of f.
d. Find the range of f.
10
Example 4
Sketch the graph of the function with the given rule. Find the domain and range of the function.
Dr. Erickson2.1 Functions and Their Graphs
2 2 1 . 1
if 0. 1
2 1 if 0
f x x f x x
x xf x x f x
x x
a. b
c d.
11
Vertical Line Test
A curve in the xy-plane is the graph of a function y = f (x) if and only if each vertical line intersects it in at most one point.
a. Function Not a function
Dr. Erickson2.1 Functions and Their Graphs
12
Example 5
Use the vertical line test to determine whether the graph represents y as a function of x.
Dr. Erickson2.1 Functions and Their Graphs
13 Example 6
Dr. Erickson2.1 Functions and Their Graphs
2.1 Functions and Their Graphs
14 Example 7
Dr. Erickson
2.1 Functions and Their Graphs
15 Example 8
Dr. Erickson
16 Next Class
We will discuss the following concepts: The Sum, Difference, Product, and Quotient of Functions
Composition of Functions
Mathematical Models
Rational and Power Functions
Economic Models
Please read through Section 2.2 – The Algebra of Functions and Section 2.3 – Function and Mathematical Models in your text book before next class.
Dr. Erickson2.1 Functions and Their Graphs