2.6 Mathematical Models: Building Functions
Building Functions
Example 1. Let P = (x, y) be a point on the graph of y = x2 8.
a) Express the distance d from P to the point (0, 1) as a function of x.
b) What is d if x = 0?
c) What is d if x = 1?
Example 2.

2.3 Properties of Functions
Recall from Section 1.2:
A graph is even if and only if (x, y) is on the graph whenever (x, y) is on the graph if and only
if the graph is symmetric with respect to the y-axis.
A graph is odd if and only if (x, y) is on the g

2.6. Mathematical Models: Building Functions
Example 1. Let P = (x, y) be a point on the graph of y = x2 8.
a) Express the distance d from P to the point (0, 1) as a function of x.
b) What is d if x = 0?
c) What is d if x = 1?
Example 2. A right triangle

2.4 Library of Functions; Piecewise-Defined Functions
Constant Function: f (x) = b, b a real number
Domain: (, )
Range: cfw_b
y-Intercept: (0, b)
The graph is a horizontal line
Even function
Identity Function: f (x) = x
Domain: (, )
Range: (, )
y-

2.1 Functions
A
second set Y .
is a correspondence between two sets, a first set X and a
If x is an element of X and y is an element of Y and if a relation exists between x and y, then we say that
to y or that y
x
x, and we write
.
A Few Ways to Represent

5.4 Logarithmic Functions
Since the exponential function y = f (x) = ax , where a > 0 and a 6= 1 is a one-to-one function, the
exponential function has an inverse function that is defined implicitly by the equation
, where a > 0 and a 6= 1, is denoted by

1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry
An
is a statement in which two expressions involving x and y are equal. The expressions are called
of the
equation. Any values of x and y that result in a true statement are said to
the equati

5.1 Composite Functions
, denoted by f g (read as
Given two functions f and g, the
), is defined by
The domain of f g is the set of all numbers x in the domain of g such that g(x) is in the domain of
f.
Another way to think of composition of functions is

3.3 Quadratic Functions and Their Properties
A
is a function of the form
where a, b, and c are real numbers and a 6= 0. The domain of a quadratic function is the set of all real
numbers.
Quadratic functions are used to model a variety of real-world applic

4.1 Polynomial Functions and Models
A polynomial function
in one variable is a function of the form
where an , an1 , ., a1 , a0 are constants called the coefficients
of the polyleading
coefficient
nomial, n 0 is an integer, and x is the variable. If an 6=

5.2 One-to-One Functions; Inverse Functions
A function is one to one
if any two different inputs in the domain correspond to two different outputs in the range. That is, if x1 and x2 are two different inputs of a function
f , then f is one-to-one if f (x1

5.4 Logarithmic Functions
Since the exponential function y = f (x) = ax , where a > 0 and a 6= 1 is a one-to-one function, the
exponential function has an inverse function that is defined implicitly by the equation
The
log
logarithmic function with base a

4.4 Polynomial and Rational Inequalities
Example 1. Solve x(x + 2)2 0 by using the graph of the function f (x) = x(x + 2)2 .
General Method for Solving Polynomial Inequalities Algebraically:
1. Write the inequality so that a polynomial expression f is on

5.2 One-to-One Functions; Inverse Functions
A function is
if any two different inputs in the domain correspond to two different outputs in the range. That is, if x1 and x2 are two different inputs of a function
f , then f is one-to-one if f (x1 ) 6= f (x2

4.5 The Real Zeros of a Polynomial Function
Goal: Find the real zeros of a polynomial function when it is not factored or cannot be easily factored.
The Division Algorithm for Polynomials: If f (x) and g(x) denote polynomial functions and if
g(x) is a pol

5.7 Financial Models
Interest is money paid for the use of money. The total amount borrowed is called the principal
The rate of interest
, expressed as a percent, is the amount charged for the
use of the principal for a given period of time, usually on a

2.5 Graphing Techniques: Transformations
Sometimes we are asked to graph a function that is similar to a graph we already know. In this
section, we look at some of these functions and develop techniques for graphing them. Collectively,
these techniques ar

4.3 The Graph of a Rational Function
Analyzing the Graph of a Rational Function:
1. Factor the numerator and denominator of R. Find the domain of the rational function.
2. Write R in lowest terms.
3. Find an plot the intercepts of the graph. Use multiplic

5.1 Composite Functions
Given two functions f and g, the
f composed with g
composite function
, denoted by f g (read as
), is defined by
The domain of f g is the set of all numbers x in the domain of g such that g(x) is in the domain of
f.
Another way to

4.2 Properties of Rational Functions
A rational function
is a function of the form
where p and q are polynomial functions and q is not the zero polynomial.
The domain of a rational function is the set of all real numbers except those for which the
denomi

Name:
Section Color:
Canvas Assignment: Module 2 Part A
You are allowed to work in groups of up to four students. You may use your notes. I am available for
assistance.
1. (1 point) State the domain and range for each relation. Determine whether each rela

History 17A: U.S. History to 1870/Fall 2016/Dr. Oliver A. Rosales
Primary Source Analysis Essay # 1 Draft Assignment
Value: 50 points
Draft Due Date: Sep 26 in class, hard copy
Assignment Learning Objectives Include:
Analysis of primary and secondary sou

2.2 The Graph of a Function
When a function is defined by an equation in x and y, the
is the graph of the equation, that is, the set of points (x, y) in the xy-plane that satisfies the equation.
Recall from the previous section that one way to represent a

3.1 Properties of Linear Functions and Linear Models
A
is a function of the form
The graph of a linear function is a line with slope m and y-intercept (0, b). Its domain is the set of all
real numbers.
.
Functions that are not linear are said to be
2
Exam

1.4 Circles
Suppose that we wish to solve the quadratic equation
x2 = p
where p 0. Then
This gives the following method.
Square Root Method: If x2 = p and p 0, then x =
p or x = p.
Example 1. Solve x2 = 16.
Completing the square is a process that takes an

4.6 Complex Zeros; Fundamental Theorem of Algebra
One property of a real number is that its square is nonnegative. For example, there is no real
number x for which x2 = 1. To remedy this situation, we introduce a new number called the
.
, which we denote

3.2 Building Linear Models from Data
A
plotted.
is a graph where the ordered pairs of a set of data are
One of our first steps after plotting the data in a scatter diagram will be to determine whether our data
set represents a linear or nonlinear relation

4.2 Properties of Rational Functions
A rational function
is a function of the form
where p and q are polynomial functions and q is not the zero polynomial.
The domain of a rational function is the set of all real numbers except those for which the
denomi

3.5 Inequalities Involving Quadratic Functions
To solve the inequality
ax^2 + bx + c > 0, a not = 0
graph the function f (x) = ax2 +bx+c and, from the graph, determine
where it is above the x-axis.
the x-values
To solve the inequality
ax^2 + bx + c < 0, a