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1.1 - Functions

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Interval Notation

Algebraic Notation Graph Interval Notation

x > 3

-1 ≤ x < 5

x < -2 or x ≥ 4

|3

|-1

|5

|4

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A function is a relationship or correspondence between two sets of numbers, in which each member of the first set (called the domain) corresponds to one an only one member of the second set (called the range).

Function, Domain, and Range

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Domain Range

X Y

f

x2

x1

x3

y2

y1

y3

5

-4

1

6

1

-8

-2

3

5

1

-2

7

-2

5

-2

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Functions? State the domain and range.

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Yes, it is a function. Each domain element corresponds to exactly one range element.

Explanations

No, it is not a function. Some domain element corresponds to more than one range value.

If asked, how do you state that a relation is or is not a function?

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Function Representation

1. Verbal or Written2. Numerically3. Graphically4. Symbolically

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Examples

1. A correspondence between the students in this class and their student identification numbers.

2. 2,3 1,3 2,5 10,5

3. 1,2 2,2 3,1 4,2

4. 0,0 1,0 3,0 5,0

Do the following relations represent functions?

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x

yFunction?

10

x

y

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Theorem Vertical Line Test

If any vertical line drawn on the graph of a relation crosses the graph more than once, the relation does not represent a function.

Caution: It is not sufficient to state that a graph represents a function because it passes the vertical line

test.

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4

0

-4(0, -3)

(2, 3)

(4, 0)(10, 0)

(1, 0)

x

y

Determine the domain, range, and intercepts of the following graph.

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Example Does this equation represent a function? Why or why not?

(x – 2)2 + (y + 4)2 = 25

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Linear Functions y = mx + b [Slope-Intercept Form of a Line]

m = slopeb = y-intercept

y – y1 = m(x – x1) – [Point-Slope Form of a Line]

m = slope(x1, y1) = point on line

y = y1 [Horizontal Line]

m = 0

x = x1 [Vertical Line]

m = undefined or none

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Linear Functions - ExamplesDetermine the equation of the line through (-2, ¾) with slope m = ½ .

Determine the equation of the line through (4, -1) with no slope.

Determine the equation of the line through (-2, 5) with m = 0.

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Example

If x = 3, then y = _____? There is only one answer to this question.

Let y = 2x – 3y = x2 + 2x – 1y = | x + 5 |

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Example

If x = 3, then h = _____? There is only one answer to this question.

Let f = 2x – 3g = x2 + 2x – 1h = | x + 5 |

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Function Notation

f (x)

f is the name of the function

x is the variable into which we substitute

values or other expressions

x is called the independent variable and f(x) is the dependent variable.

Does Not Mean f times

x.

Read: “f of x”

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Find the domain of the following functions:

(a)

(b)

12)( xxf

1)(

x

xxg

(c) xxh 4)(

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Evaluating Functions

Let f(x)=2x – 3 and g(m) = | m2 – 2m + 1|. Determine:

(a) f(-3)(b) g(-2)(c) f(a)(d) g(a + 1)

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The Difference Quotient

h

xfhxf )()(

Example

If f (x) = x2 – 3, determine the difference quotient.

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= (x2 + 2hx + h2) – 3

= x2 + 2hx + h2 – 3

First, we need to determine f (x + h)

f (x + h) = ( )2 – 3x + h

f (x) = x2 – 3

The Difference Quotient

x

hxh

hxh

h

hhx

h

xhhxx

h

xfhxf

2)2(2

]3[]32[

)()(

2

222

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The Difference Quotient

1. 2

2.5

f x x

xg x

x

Example

Determine the difference quotient for each of the following and simplify.

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Increasing and Decreasing Functions

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 we have f(x1) < f(x2).

A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 we have f(x1) > f(x2).

A function f is constant on an open interval I if, for all choices of x, the values f(x) are equal.

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4

0

-4 (0, -3)

(2, 3)

(4, 0)

(10, -3)

(1, 0)

x

y

(7, -3)

Increasing, Decreasing Constant, Local Maximum, Local Minimum

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When functions are defined by more than one equation, they are called piece-wise defined functions.

Piece Wise Defined Functions

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For the following piece-wise defined function:

a) Find f (-1), f (1), f (3).b) Sketch a graph of f.c) Find the domain of f.

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13

123

)(

xifx

xif

xifx

xf

Example

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Absolute Value

0

0

aifa

aifaa

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Example

Use the definition to do the following:

(a) Explain why | -2 | = 2.(b) Determine the exact value of | 3 – π |.(c) Write | x – 2 | as a piece wise defined

function without absolute value bars.

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Modeling With Functions

Example 1 Express the surface area of a cube as a function of its volume.

Example 2 A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.

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Odd or Even FunctionA function is odd if f (-x) = -f (x). Odd functions exhibit origin symmetry on their graphs. This means if you turn the graph upside down, it will look the same.

A function is even if f (-x) = f (x). The graphs of even functions will be symmetric to the y-axis.

Simply substitute -x for each x in the function and determine if you get f (x) or -f (x). If you get neither, it is neither odd nor even.

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Odd or Even Function

Determine if is odd or even.

Determine if is odd or even. xx

xxg

5

23

2

22 xxf