Ch3.1 – 3.2 Functions and Graphs

The table above establishes a relation between the year and the cost of tuition at a public college. For each year there is a cost, forming a set of ordered pairs.A relation is a set of ordered pairs (x, y). The relation above can be writtenas 4 ordered pairs as follows:

S = {(1997, 3111), (1998, 3247), (1999, 3356), (2000, 3510)} x y x y x y x y

Domain – the set of all x-values. D = {1977, 1998, 1999, 2000}

Range – the set of all y-values. R = {3111, 3247, 3356, 3510}

Year

Cost

1997

$3111

1998

$3247

1999

$3356

2000

$3510

independent variable (x)

dependent variable (y)

The cost depends on the year.

Year(x) Cost(y)1997 31111998 32471999 33562000 3510Thinking Exercise: Draw a ‘line’ in the x/y axes.

What is the Domain & Range?

Functions & Linear Data Modelingy – Profit in thousands of $$(Dependent Var)

x - Years in business (Independent Var)

(0,-3)

(6,0)

y intercept x

intercept

Equation: y = ½ x – 3

Function: f(x) = ½ x – 3

x y = f(x)0 -3 f(0) = ½(0)-3=-32 -2 f(2) = ½(2)-3=-26 0 f(6) = ½(6)-3=08 1 f(8) = ½(8)-3=1

Inputx

Functionf

Outputy=f(x)

A function has exactly one output value (y)for each valid input (x).

Use the vertical line test to see if an equation is a function. •If it touches 1 point at a time then FUNCTION•If it touches more than 1 point at a time then NOT A FUNCTION.

Diagrams of Functions

Function: f(x) = ½ x – 3

x y = f(x)0 -3 f(0) = ½(0)-3=-32 -2 f(2) = ½(2)-3=-26 0 f(6) = ½(6)-3=08 1 f(8) = ½(8)-3=1

0

2

6

-3

8

-2

01

f

12

3

f

A function NOT a function

4

5

45

6

A function is a correspondence fro the domain to the range such that each elementin the domain corresponds to exactly one element in the range.

How to Determine if an equation is a function

Example 1: x2 + y = 4

y = 4 – x2

For every value of x thereIs exactly 1 value for y, so This equation IS A FUNCTION.

Graphically: Use the vertical line test

Symbolically/Algebraically: Solve for y to see if there is only 1 y-value.

Example 2: x2 + y2 = 4

y2 = 4 – x2

y = 4 – x2 or y = - 4 – x2

For every value of x thereare 2 possible values for y, so This equation IS NOT A FUNCTION.

Are these graphs functions?

Use the vertical line test to tell if the following are functions:

y = x3

Origin Symmetry

y = x2 Y-axisSymmetry

x = y2

X-axis Symmetry

More on Evaluation of Functions

f(x) = x2 + 3x + 5

Evaluate: f(2)f(2) = (2)2 + 3(2) + 5f(2) = 4 + 6 + 5f(2) = 15

Evaluate: f(x + 3)f(x + 3) = (x + 3)2 + 3 (x + 3) + 5f(x + 3) = (x + 3)(x + 3) + 3x + 9 + 5f(x + 3) = (x2 + 3x + 3x + 9) + 3x + 14f(x + 3) = (x2 + 6x + 9) + 3x + 14f(x + 3) = x2 + 9x + 23

Evaluate: f(-x)f (-x) = ( -x)2 + 3( -x) + 5 f (-x) = x2 - 3x + 5

More on Domain of FunctionsA function’s domain is the largest set of real numbers for which the value f(x)is a real number. So, a function’s domain is the set of all real numbers MINUS the following conditions:

• specific conditions/restrictions placed on the function

• Bounds relating to real-life data modeling (Example: y = 7x, where y is dog years and x is dog’s age)

• values that cause division by zero

• values that result in an even root of a negative number

What is the domain the following functions:

1. f(x) = 6x 2. g(x) = 3x + 12 3. h(x) = 2x + 1 x2 – 9

Definition of a Difference Quotient

The expression below is called the difference quotient.(This expression will become useful later, so…….Stay Tuned ….)

Example function: f(x) = x2 + 3x + 5

f(x + h) = (x + h )2 + 3( x + h ) + 5 = (x + h)(x + h) + 3x + 3h + 5 = x2 + 2xh + h2 + 3x + 3h + 5

Challenge yourself!For the same function f(x), find:

Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows:

Sum: (f + g)(x) = f (x)+g(x)Difference: (f – g)(x) = f (x) – g(x)Product: (f • g)(x) = f (x) • g(x)Quotient: (f / g)(x) = f (x)/g(x), provided g(x) does not equal 0

Example: Let f(x) = 2x+1 and g(x) = x2-2.

f+g = 2x+1 + x2-2 = x2+2x-1

f-g = (2x+1) - (x2-2)= -x2+2x+3

fg = (2x+1)(x2-2) = 2x3+x2-4x-2

f/g = (2x+1)/(x2-2)

Adding & Subtracting FunctionsIf f(x) and g(x) are functions, then:

(f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x)

Examples: f(x) = 2x + 1 and g(x) = -3x – 7 Method1 Method1 (f + g)(4) = 2(4) + 1 + -3(4) – 7 (f – g)(6) = 2(6) + 1 – [-3(6) – 7] = 8 + 1 + -12 – 7 = 12 + 1 - [-18 – 7] = 9 + -19 = 13 - [-25] = -10 = 13 + 25 Method2 Method2 = 38 (f + g)(4) = 2x + 1 + -3x – 7 (f + g)(6) = 2x + 1 - [-3x – 7] = -x – 6 = 2x + 1 + 3x + 7 = -4 – 6 = 5x + 8 = - 10 = 5(6) + 8

= 30 + 8 = 38

Adding/subtracting also extends to non-linearfunctions you will see in a subsequent chapter.

Function Practice

1) y = 9 Domain? = _____________ 3 – 8x

2) X = y2 (Is y a function of x?) _______________

3)Y = x9 (Is y a function? ________ Domain? _________)

4)F(x) = x2 – 3x + 7 and g(x) = -3x2 -7x + 7

(f + g)(x) = _____________________________

(f – g)(x) = _____________________________

More Practice

f(x) = 2x2

x4 + 1

(a)Is the point (-1, 1) on the graph of f?

(b)If x = 2, what is f(x)? What point is on the graph of f?

(c)If f(x) = 1, what is x? What point(s) are on the graph of f?

(d)What is the domain of f?

(e)List the x-intercepts, if any, of the graph of f.

(f)List the y-intercept, if there is one, of the graph of f.

Application: GolfA golf ball is hit with an initial velocity of 130 feet per second at an inclination of 45 degrees to the horizontal. In physica, it is established that the height h of the golf ball is given by the function:

h(x) = -32x2 / 1302 + x

Where x is horizontal distance that the golf ball has traveled.

(a)Determine the height of the golf ball after it has traveled 100 feet

(b)What is the height after it has traveled 300 feet?

(c)What is the height after it has travelled 500 feet?

(d)How far was the golf ball hit?

(e)Use a Ti-84 to graph the function h=h(x)

(f)Use a Ti-84 to determine the distance that the ball has traveled when the hieght of the ball is 90 feet.

(g)Create a TABLE with TblStart = 0 and ΔTbl = 25. To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height?

(h)Adjust the value of ΔTbl until you determine the distance, to within 1 foot, that the ball travels before it reaches a maximum height.

3.3 Even/Odd Functions RevisitedY-Axis Symmetry even functions f (-x) = f (x) For every point (x,y), the point (-x, y) is also on the graph.Test for symmetry: Replace x by –x in equation. Check for equivalent equation.

Origin Symmetry odd functions f (-x) = -f (x)For every point (x, y), the point (-x, -y) is also on the graph.Test for symmetry: Replace x by –x , y by –y in equation. Check for equivalent equation.

y = x3

ODD

y = x2 EVEN

Test-y = (-x)3

-y = -x3

y = x3

Testy = (-x)2

y = x2

Try these withoutUsing a graph:

y = 3x2 – 2

y = x2 + 2x + 1

Increasing, Decreasing, and Constant Functions

Constantf (x1) = f (x2)

(x1, f (x1))

(x2, f (x2))

Increasingf (x1) < f (x2)

(x1, f (x1))

(x2, f (x2))

Decreasingf (x1) > f (x2)

(x1, f (x1))

(x2, f (x2))

A function is increasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) < f (x2).

A function is decreasing on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) > f (x2).

A function is constant on an interval if for any x1, and x2 in the interval, where x1 < x2, then f (x1) = f (x2).

Observations• Decreasing on the

interval (-oo, 0)

• Increasing on the interval (0, 2)

• Decreasing on the interval (2, oo).

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

5

4

3

1

-1-2

-3

-4-5

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

5

4

3

2

1

-1-2

-3

-4-5

a.

b.

More Examples

Observationsa. Two pieces (a piecewise function)

b. Constant on the interval (-oo, 0).

c. Increasing on the interval (0, oo).

Challenge Yourself: What might bethe definition of the piecewise functionfor this graph? (You will learn about theseLater. Can you guess what it might be?)

Relative (local) Min & Max

0

x

y

180 36090 270

1

-1

f(x) = sin (x) x y0 0/2 1 03/2 -12 02

-2

The point at which a function changes its increasing or decreasing behavior is called a relative minimum or relative maximum.

(90, f(90)) f(90), or 1, is a local max

(270, f(270))f(270), or -1, is a local min

A function value f(a) is a relative maximum of f if there exists an open interval about a such that f(a) > f(x) for all x in the open interval.

A function value f(b) is a relative minimum of f if there exists an open interval about b such that f(b) < f(x) for all x in the open interval.

Slope & Average Rate of Changey = x2 - 4x + 4

y - $$ in thousands

xYrs

(0,-3)

(6,0)

y = ½ x – 3

The slope of a line may beinterpreted as the rate of change.The rate of change for a line is constant (the same for any 2 points) y2 – y1

x2 – x1

Non-linear equations do not have a constant rate of change. But you canFind the average rate of change fromx1 to x2 along a secant to the graph. f(x2) – f(x1) x2 – x1

See Page 236 for more examples.

Application: Medicine

The concentration C of a medication in the bloodstream t hours after being administered is given by:

C(t) = -.002x4 + .039t3 - .285t2 + .766t + .085

(a)After how many hours will the concentration be highest?

(b)A woman nursing a child must wait until the concentration is below .5 before she can feed him. After taking the medication, how long must she wait before feeding her child?

3.4 Library of Functions/Common Graphsy = c

x

y = x

x

y = x2

x

y = x3

x

y = x

x

y = |x|

x

Can you graph : y = ½ (x + 2)3 + 2

y = 1/x y = x1/3

x

Step Function Application Example

y = int(x) or y = [[x]] (Greatest Integer Function)

f(x) = int(x) y – Tax (+) or Refund (-) in thousands of $$

x – Income in $10,000’s

Find:1) f (1.06)2) f (1/3)3) f (-2.3)

• What other applications of the step function can you think of?

Piecewise Functions

f(x) = x2 + 3 if x < 0 5x + 3 if x>=0

f(-5) = (-5)2 + 3 = 25 + 3 = 28

f(6) = 5(6) + 3 = 33

A function that is defined by two (or more) equations over a specified domain is called a piecewise function.

See Page 247 for more examples

3.5 Transformation of FunctionsA transformation of a graph is a change in its position, shape or size.

For a given function, y = f(x)

y = f(x) +c [shift up c]y = f(x) – c [shift down c]

y = f(x + c) [shift left c]y = f(x – c) [shift right c]

y = -f(x) [flip over x-axis]y = f(-x) [flip over y-axis]

y = cf(x) [multiply y value by c] [if c > 1, stretch vertically] [if 0 < c < 1, shrink vertically]

Example function: y = x2

Graph: y = x2 + 4y = x2 - 4y = (x+4)2

y = (x – 4)2

y = -x2

y = (-x)2

y = ½ x2

More Transformation Practice

Suppose that the x-intercepts of the graph of y = f(x) are -5 and 3

(a)What are the x-intercepts of the graph of y = f(x + 2)

(b)What are the x-intercepts of the graph of y = f(x – 2)

(c)What are the x-intercepts of the graph y = 4f(x)

(d)What are the x-intercepts of the graph of y = f(-x)

A)

B)

C)

D)

3.6 Application: Bob wants to fence in a rectangular garden in his yard. He has 62 feet of fencing to work with and wants to use it all. If the garden is to be x feet wide, express the area of the garden as a function of x.