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

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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