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Chapter 2
Functions and
Graphs
Section 1 - Functions
Section 2 - Elementary Functions: Graphs
& Transformations
Section 3 - Quadratic Functions
Section 4 - Polynomial & Rational Functions
Section 5 - Exponential Functions
Section 6 - Logarithmic Functions
Section 1
Functions
Learning Objectives for Section
2.1
Functions
 The student will be able to do point-by-point plotting of
equations in two variables.
 The student will be able to give and apply the definition of
a function.
 The student will be able to identify domain and range of a
function.
 The student will be able to use function notation.
 The student will be able to solve applications.
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Table of Content





Equations in Two Variables
Definition of a Function
Functions Specified by Equations
Function Notation
Applications
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Terms








point-by-point plotting
function
domain
range
independent variable
dependent variable
linear function
constant function
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





break-even analysis
profit-loss analysis
revenue function
loss function
profit function
cost fixed/variable
price-demand function
4
Equations in Two Variables
Graphing an Equation
 To sketch the graph an equation in x and y, we need to find
ordered pairs that solve the equation and plot the ordered
pairs on a grid. This process is called point-by-point
plotting.
For example, let’s plot the graph of the equation
y=
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x +
2
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Graphing an Equation:
Making a Table of Ordered Pairs
 Make a table of ordered
pairs that satisfy the
equation
x
–3
y = x2 + 2
–2
y
(–3)2+2 =
11
(–2)2+2 = 6
–1
0
1
2
(–1)2+2 = 6
(0)2+2 = 2
(1)2+2 = 3
(2)2+2 = 6
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Graphing an Equation:
Plotting the points
 Next, plot the points and connect them with a smooth
curve. You may need to plot additional points to see the
pattern formed.
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Functions
The previous graph is the graph of a function. The idea of a
function is this:
 A correspondence between two sets D and R such that to
each element of the first set, D, there corresponds one
and only one element of the second set, R.
 The first set is called the domain, and the set of
corresponding elements in the second set is called the
range.
For example, the cost of a pizza (C) is related to the size of
the pizza. A 10 inch diameter pizza costs $9.00, while a 16
inch diameter pizza costs $12.00.
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Function Definition
 You can visualize a function by the following diagram which
shows a correspondence between two sets: D, the domain of
the function, gives the diameter of pizzas, and R, the range of
the function gives the cost of the pizza.
10
12
16
9.00
10.00
12.00
domain D
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range R
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Function Definition
TABLA 1
Dominio
Rango
Dominio
Rango
Numero
Cubo
Numero
Cuadrado
-2
TABLA 3
TABLA 2
-8
Dominio
Rango
Numero
Raiz Cuadrada
-2
0
-1
-1
-1
4
1
0
0
0
1
4
1
1
1
0
9
2
8
0
1
-1
2
-2
3
-3
2
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Functions Specified by Equations
 If in an equation in two variables, we get exactly one
output (value for the dependent variable) for each input
(value for the independent variable), then the equation
specifies a function. The graph of such a function is just
the graph of the specifying equation.
 If we get more than one output for a given input, the
equation does not specify a function.
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Functions Specified by Equations
 Consider the equation that was graphed on a previous slide
y = x2 + 2
(–2,2) is an
ordered pair of
the function.
–2
(-2)2 + 2
6
Barnett/Ziegler/Byleen College Mathematics 12e
Input:
x = –2
Process:
square (–2),
then add 2
Output:
result is 6
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Example 2
Functions Specified by Equations
 Determine which of the following equations specify
functions with inpependent variable x.
• (A) 4y – 3x = 8
• (B) y2 – x2 = 9
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Example 2
Functions Specified by Equations
Solution (A)
Since each input value x
corresponds to exactly
one output value. this
equation specify a
function.
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Example 2
Functions Specified by Equations
Solution (B)
Since
is always a positive
real number, for any value of x, and
since each positive number has two
square roots, then to each input value
x there corresponds two output
values y. So, this equation does not
specify a function.
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Vertical Line Test for a Function
If you have the graph of an equation, there is an easy
way to determine if it is the graph of an function. It is
called the vertical line test which states that:
An equation specifies a function if each vertical line in
the coordinate system passes through at most one
point on the graph of the equation.
If any vertical line passes through two or more points
on the graph of an equation, then the equation does
not specify a function.
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Vertical Line Test for a Function
(continued)
This graph is not the graph of a
function because you can draw a
vertical line which crosses it
twice.
This is the graph of a
function because any vertical
line crosses only once.
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Vertical Line Test for a Function
(continued)
The vertical test implies that
equations of the form y = mx + b
specify functions; they are called
linear functions.
Similarly, equations of the form
y = b specify functions; they are
called constant functions, and
their graphs are horizontal lines.
The vertical-line test implies that
equations of the form x = a do
not specify functions, the graph is
a vertical line.
Barnett/Ziegler/Byleen College Mathematics12e
y
x
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Function Notation
 The following notation is used to describe functions. The
variable y will now be called f (x).
 This is read as “ f of x” and simply means the y coordinate
of the function corresponding to a given x value.
Our previous equation
y = x2 + 2
can now be expressed as
f(x) = x2 + 2
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Function Evaluation
 Consider our function
f(x) = x2 + 2
 What does f (–3) mean?
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Function Evaluation
 Consider our function
f(x) = x2 + 2
 What does f (–3) mean?
Replace x with the value –3 and evaluate the expression
f (3)  (3)2  2
The result is 11 . This means that the point (–3,11) is on
the graph of the function.
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Some Examples
 1.
f (x)  3x  2
f (2)  3(2)  2  4  2
f (a)  3(a)  2
f (6  h)  3(6  h)  2  18  3h  2
 16  3h
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Domain of a Function
 Consider
f ( x)  3 x  2
f (0)  ?
f (0)  3(0)  2  2
which is not a real number.
 Question: for what values of x is the function defined?
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Domain of a Function
 Answer:
f ( x)  3 x  2
is defined only when the radicand (3x – 2) is equal to
or greater than zero. This implies that
2
x
3
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Domain of a Function
(continued)
 Therefore, the domain of our function is the set of real
numbers that are greater than or equal to 2/3.
 Example: Find the domain of the function
1
f ( x) 
x4
2
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Domain of a Function
(continued)
 Therefore, the domain of our function is the set of real
numbers that are greater than or equal to 2/3.
 Example: Find the domain of the function
1
f ( x) 
x4
2
 Answer:
x x  8 , [8, )
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Domain of a Function:
Another Example
 Find the domain of
1
f ( x) 
3x  5
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Domain of a Function:
Another Example
 Find the domain of
1
f ( x) 
3x  5
 In this case, the function is defined for all values of x
except where the denominator of the fraction is zero. This
means all real numbers x except 5/3.
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Function Notation: Example
 Using function notation. Find for




(A)
(B)
(C)
(D)
f(a)
f(a + h)
f(a + h) – f(a)
[f(a + h) – f(a)] / h h ≠ 0
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Function Notation: Example
 Solution.
(A)
(B)
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Function Notation: Example
 Solution.
(C)
(D)
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Break-Even and Profit-Loss
Analysis
Applications
 Any manufacturing company has costs C and revenues R.
 The company will have a loss if R < C, will break even
if R = C, and will have a profit if R > C.
 Costs include fixed costs such as plant overhead, etc. and
variable costs, which are dependent on the number of
items produced.
C = a + bx
(x is the number of items produced)
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Break-Even and Profit-Loss
Analysis (continued)
 Price-demand functions, usually determined by financial
departments, play an important role in profit-loss analysis.
p = m – nx
(x is the number of items than can be sold at $p per item.)
 The revenue function is
R = (number of items sold) ∙ (price per item)
= xp = x(m – nx)
 The profit function is
P = R – C = x(m – nx) – (a + bx)
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Mathematical Modeling 1
The price-demand function for a company is given by
p( x)  1000  5x,
0  x  100
where x represents the number of items and p(x) represents the
price of the item. Determine the revenue function and find the
revenue generated if 50 items are sold.
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Solution
Revenue = Price ∙ Quantity, so
R(x)= p(x) ∙ x = (1000 – 5x) ∙ x
When 50 items are sold, x = 50, so we will evaluate the
revenue function at x = 50:
R(50)  (1000  5(50)) 50  37,500
The domain of the function has already been specified. We
are told that
0  x  100
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Example of Profit-Loss Analysis
A company manufactures notebook computers. Its
marketing research department has determined that the
data is modeled by the price-demand function
p(x) = 2,000 – 60x, when 1 < x < 25, (x is in thousands).
What is the company’s revenue function and what is its
domain?
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Answer to Revenue Problem
Since Revenue = Price ∙ Quantity,
R( x)  x  p( x)  x  (2000  60 x)  2000 x  60 x 2
The domain of this function is the same as the domain
of the price-demand function, which is 1 ≤ x ≤ 25 (in
thousands.)
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Profit Problem
The financial department for the company in the preceding
problem has established the following cost function for
producing and selling x thousand notebook computers:
C(x) = 4,000 + 500x
(x is in thousand dollars).
Write a profit function for producing and selling x thousand
notebook computers, and indicate the domain of this function.
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Answer to Profit Problem
Since Profit = Revenue – Cost, and our revenue function
from the preceding problem was R(x) = 2000x – 60x2,
P(x) = R(x) – C(x) = 2000x – 60x2 – (4000 + 500x)
= –60x2 + 1500x – 4000.
The domain of this function is the same as the domain of
the original price-demand function, 1< x < 25 (in
5000
thousands.)
Thousand dollars
Thousand cameras
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Mathematical Modeling 2
Price-Demand and Revenue Modeling.
A manufacturer of a popular digital
camera wholesales the camera to retail
outlets throughout the USA. Using
statistical methods, the financial
department in the company produces
the price-demand data in Table A, where
p is the wholesale price per camera at
which x million cameras are sold.
Table A. Price-Demand
x (Millions)
P ($)
2
87
5
68
8
53
12
37
The following price-demand function was obtained:
p(x) = 94.8 – 5x
1 <= x <= 15
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Mathematical Modeling 2
Price-Demand and Revenue Modeling (Continued).
A. Plot the data in Table A. Then sketch a graph of the pricedemand function in the same coordinate system.
B. What is the company’ s revenue function for this camera and
what is its domain?
C. Complete Table B, computing revenues to the nearest million
dollars.
D. Plot the data in Table B. Then sketch a graph of the revenue
function using this points.
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Mathematical Modeling 2
Price-Demand and Revenue Modeling (Continued).
Table B. Revenue
x (Millions)
R(x) (Million $)
1
90
3
6
9
12
15
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Solution to Mathematical Modeling 2
A.
B. Revenue = Price ∙ Quantity, so
R(x)= p(x) ∙ x = (94.8 – 5x) ∙ x
Domain: 1 <= x <= 15
y
.
.
.
.
x
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Solution to Mathematical Modeling 2
C.
D.
y
500
Table B. Revenue
450
x (Millions)
R(x) (Million $)
1
90
350
3
239
250
6
389
9
448
400
300
200
150
100
50
12
418
0
1
15
2
3
4
5
6
x
297
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Chapter 2
Functions and
Graphs
Section 1
Functions
END
Last Update: February 14/2013