2.1 Functions and Their GraphsByDr. Julia Arnold

ByDr. Julia Arnold

A function is a rule that assigns to each element in a set A one and only one element in a set B.

Domain-All elements in domain are assigned to something

Range- The range consists of the elements used by the domain. The Range

In general our domain will begin with the Real Numbers.However, there are some equations which require us to use a subset of the Reals for the domain.These equations are:1. Certain types of word problems which pertain to measurable items. For example Volume of a box in terms of the size of material.

2.Equations where the variable is in the denominator of a fraction:

Equations which contain the variable under a radical:

5x2

y

7xy

Or a combination of the above.

y = x y = x3

y = x2

Continued

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

In pre-calculus you studied the graphs of some common functions

Functions continued:

y = y =

y =

x

x1

x

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

The Vertical-Line Test shows you which graphs are functions: If you pass a vertical line across the graph it should only intersect the graph one point at a time.Non-functions:

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

Problem

Find the domain of the function: ))(()(

3x2x1x

xf

This problem has both a radical and a fraction.We must find the numbers which keep the radicand positive and the denominator non-zero.

Solution: 3x2x , or the denominator would be zero.

In order that the radicand is positive or zero

1x

01x

-1-2 0 1 2 3

Since the domain must be greater than or equal to one, we don’t have to be concerned with -2. However, 3 is greater than 1 but must not be in the domain.So the domain is

,, 331

Sketch the graph of the function with the given rule. Find the domain and range of the function.

1ifx1x

1x1if0

1ifx1x

xf

,

,

,

)(This is called a piece-wise function. It has 3 pieces.

The domain is represented by the 3 if statements:x < -1 ,x > 1 which when put together is all real numbers.

1x1

The first and last equation will be slanted lines.The middle equation is a horizontal line.

Problem 46Sketch the graph of the function with the given rule. Find the domain and range of the function.

1ifx1x

1x1if0

1ifx1x

xf

,

,

,

)(

-4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

x

y

The range is the set of numbersused in the graph forthe y value.

0y