1. Ch3.1 – 3.2 Functions and Graphs
Year
1997
1998
independent variable (x)
1999
2000
$3111
$3247
$3356
$3510
dependent variable (y)
Cost
The cost depends on the year.
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 written
as 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(x) Cost(y)
Thinking Exercise: Draw a ‘line’ in the x/y axes.
What is the Domain & Range?

2. Functions & Linear Data Modeling
Input x
Functions & Linear Data Modeling
y – Profit in thousands of $$
(Dependent Var)
x - Years in business
(Independent Var)
Function f
(6,0)
Output
y=f(x) y
intercept x
intercept
(0,-3)
Equation: y = ½ x – 3
Function: f(x) = ½ x – 3
x y = f(x)
f(0) = ½(0)-3=-3
f(2) = ½(2)-3=-2
f(6) = ½(6)-3=0
f(8) = ½(8)-3=1
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.

3. Diagrams of Functions f -3 -2 6 8 1 1 2 3
A function is a correspondence fro the domain to the range such that each element
in the domain corresponds to exactly one element in the range. f
Function: f(x) = ½ x – 3
x y = f(x)
f(0) = ½(0)-3=-3
f(2) = ½(2)-3=-2
f(6) = ½(6)-3=0
f(8) = ½(8)-3=1 -3 2 -2 6 8 1 f 4 5 6 1 2 3 4 5
A function
NOT a function

4. How to Determine if an 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 1: x2 + y = 4
y = 4 – x2
For every value of x there
Is exactly 1 value for y, so
This equation IS A FUNCTION.
Example 2: x2 + y2 = 4
y2 = 4 – x2
y = 4 – x2 or y = – x2
For every value of x there
are 2 possible values for y, so
This equation IS NOT A
FUNCTION.

5. Are these graphs functions?
Use the vertical line test to tell if the following are functions:
y = x2
Y-axis
Symmetry
x = y2
X-axis
Symmetry
y = x3
Origin
Symmetry

6. More on Evaluation of Functions
f(x) = x2 + 3x + 5
Evaluate: f(2)
f(2) = (2)2 + 3(2) + 5
f(2) =
f(2) = 15
Evaluate: f(x + 3)
f(x + 3) = (x + 3)2 + 3 (x + 3) + 5
f(x + 3) = (x + 3)(x + 3) + 3x
f(x + 3) = (x2 + 3x + 3x + 9) + 3x + 14
f(x + 3) = (x2 + 6x + 9) + 3x + 14
f(x + 3) = x2 + 9x + 23
Evaluate: f(-x)
f (-x) = ( -x)2 + 3( -x) + 5
f (-x) = x2 - 3x + 5

7. More on Domain of Functions
A 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:
f(x) = 6x 2. g(x) = x h(x) = 2x + 1
x2 – 9

8. 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:

9. 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)

10. Adding & Subtracting Functions
If 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 and g(x) = -3x – 7
Method Method1
(f + g)(4) = 2(4) (4) – (f – g)(6) = 2(6) + 1 – [-3(6) – 7]
= – = [-18 – 7]
= = [-25]
= =
Method Method2 =
(f + g)(4) = 2x x – (f + g)(6) = 2x [-3x – 7]
= x – = 2x x + 7
= – = 5x + 8
= = 5(6) + 8 = =
Adding/subtracting also extends to non-linear
functions you will see in a subsequent chapter.

11. Function Practice 1) y = 9 Domain? = _____________ 3 – 8x
2) X = y2 (Is y a function of x?) _______________
Y = x9 (Is y a function? ________ Domain? _________)
F(x) = x2 – 3x and g(x) = -3x2 -7x + 7
(f + g)(x) = _____________________________
(f – g)(x) = _____________________________

12. More Practice f(x) = 2x2 x4 + 1
Is the point (-1, 1) on the graph of f?
If x = 2, what is f(x)? What point is on the graph of f?
If f(x) = 1, what is x? What point(s) are on the graph of f?
What is the domain of f?
List the x-intercepts, if any, of the graph of f.
List the y-intercept, if there is one, of the graph of f.

13. Application: Golf
A 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 / x
Where x is horizontal distance that the golf ball has traveled.
Determine the height of the golf ball after it has traveled 100 feet
What is the height after it has traveled 300 feet?
What is the height after it has travelled 500 feet?
How far was the golf ball hit?
Use a Ti-84 to graph the function h=h(x)
Use a Ti-84 to determine the distance that the ball has traveled when the hieght of the ball is 90 feet.
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?
Adjust the value of ΔTbl until you determine the distance, to within 1 foot, that the ball travels before it reaches a maximum height.

14. 3.3 Even/Odd Functions Revisited
Y-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 = x2
EVEN
y = x3
ODD
Try these without
Using a graph:
y = 3x2 – 2
y = x2 + 2x + 1
Test
-y = (-x)3
-y = -x3
y = x3
Test
y = (-x)2
y = x2

15. Increasing, Decreasing, and Constant Functions
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).
Constant
f (x1) = f (x2)
(x1, f (x1))
(x2, f (x2))
Increasing
f (x1) < f (x2)
Decreasing
f (x1) > f (x2)

16. More Examples a. b. Observations Decreasing on the interval (-oo, 0)-5 -4 -3 -2 -1 1 2 3 4 5 a. b.
Observations
Decreasing on the interval (-oo, 0)
Increasing on the interval (0, 2)
Decreasing on the interval (2, oo).
Observations
a. Two pieces (a piecewise function)
b. Constant on the interval (-oo, 0).
c. Increasing on the interval (0, oo).
Challenge Yourself: What might be
the definition of the piecewise function
for this graph? (You will learn about these
Later. Can you guess what it might be?)

17. Relative (local) Min & Max
f(x) = sin (x)
x y /
3/ 2
The point at which a function changes its increasing or decreasing
behavior is called a relative minimum or relative maximum. y 2
(90, f(90)) f(90), or 1, is a local max 1 x
180
360 90
270 -1
(270, f(270))
f(270), or -1, is a local min -2
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.

18. Slope & Average Rate of Change
y - $$ in thousands x
Yrs
y = x2 - 4x + 4
y = ½ x – 3
(6,0)
(0,-3)
Non-linear equations do not have a constant rate of change. But you can
Find the average rate of change from
x1 to x2 along a secant to the graph.
f(x2) – f(x1)
x2 – x1
The slope of a line may be
interpreted as the rate of change.
The rate of change for a line is constant (the same for any 2 points)
y2 – y1
x2 – x1
See Page 236 for more examples.

19. Application: Medicine
The concentration C of a medication in the bloodstream t hours after being administered is given by:
C(t) = -.002x t t t
After how many hours will the concentration be highest?
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?

20. 3.4 Library of Functions/Common Graphs
y = c x
y = x x
y = x2 x
y = x3 x
y = x x
y = |x| x
y = 1/x
y = x1/3 x
Can you graph : y = ½ (x + 2)3 + 2

21. 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:
f (1.06)
f (1/3)
f (-2.3)
What other applications of the step function can you think of?

22. Piecewise Functions
A function that is defined by two (or more) equations over
a specified domain is called a piecewise function.
f(x) = x if x < 0
5x if x>=0
f(-5) = (-5) = = 28
f(6) = 5(6) + 3 = 33
See Page 247 for more examples

23. 3.5 Transformation of Functions
A 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 + 4
y = x2 - 4
y = (x+4)2
y = (x – 4)2
y = -x2
y = (-x)2
y = ½ x2

24. More Transformation Practice
Suppose that the x-intercepts of the graph of y = f(x) are -5 and 3
What are the x-intercepts of the graph of y = f(x + 2)
What are the x-intercepts of the graph of y = f(x – 2)
What are the x-intercepts of the graph y = 4f(x)
What are the x-intercepts of the graph of y = f(-x)

25. 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. A) B) C) D)