L21
5.1 Inverse Functions
In 3.7 we discussed algebra of functions. We know that addition and subtraction undo each other as well as multiplication and division. For some functions, called inverses, the composition can be undone as well. Recall: A functio
Exam
Name_
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1) A demand curve shows the relationship between
A) the price of a produce and the demand for the product.
B) the price of a product and the
Econ 202 Exam 1 Practice Problems
Principles of Microeconomics
Dr. Phillip Miller
Multiple Choice
Identify the choice that best completes the statement or answers the question.
Chapters 1 and 2
_
1. For an economist, the idea of making assumptions is rega
L25
5.6 Exponential Growth or Decay
In many situations, the quantity changes at a rate
proportional to the amount present.
For example,
P = 80,000e0.023t
could represent the population of Gainesville t years
from 1990. At this rate we could estimate the
L25
5.6 Exponential Growth or Decay
In many situations, the quantity changes at a rate
proportional to the amount present.
For example,
P = 80,000e0.023t
could represent the population of Gainesville t years
from 1990. At this rate we could estimate the
L24
5.4 Evaluating Logs and Changing Base
Calculators can be used to evaluate base e or base 10
logarithms.
Notation:
log10 x = log x
Example: We use a calculator to find
log 142 2.1523
ln10 2.3026
Example: Calculate log 3 5 .
Let's change log a x ln x .
L24
5.4 Evaluating Logs and Changing Base
Calculators can be used to evaluate base e or base 10
logarithms.
Notation:
log10 x = log x
Example: We use a calculator to find
log 142 2.1523
ln10 2.3026
Example: Calculate log 3 5 .
Let's change log a x ln x .
L23
5.3 Logarithmic Functions
The exponential function f ( x) = a x ( a > 0, a 1) is a
1-1 function and, therefore, it has the inverse f 1 ( x) .
We denote the inverse log a x and read logarithm to
the base a of x.
Sketch the graphs of y = 2 x and y = log
L23
5.3 Logarithmic Functions
The exponential function f ( x) = a x ( a > 0, a 1) is a
1-1 function and, therefore, it has the inverse f 1 ( x) .
We denote the inverse log a x and read logarithm to
the base a of x.
Sketch the graphs of y = 2 x and y = log
L22
5.2 Exponential Functions
Definition. The exponential function with base a is a
function of the form
f ( x) = a x ,
where a > 0 and a 1.
Note: f (0) = a 0 = _
So, the point _ is on the graph of f ( x) .
Reviewing some Rules:
Example. Write each expres
L22
5.2 Exponential Functions
Definition. The exponential function with base a is a
function of the form
f ( x) = a x ,
where a > 0 and a 1.
Note: f (0) = a 0 = _
So, the point _ is on the graph of f ( x) .
Reviewing some Rules:
Example. Write each expres
L21
5.1 Inverse Functions
In 3.7 we discussed algebra of functions. We know
that addition and subtraction undo each other as well as
multiplication and division. For some functions, called
inverses, the composition can be undone as well.
Example: Determin
L20
4.5 Rational Functions
*See Appendix for important graphs you need to
memorize*
Example. State the domain and sketch the graph of the
1
.
function y =
x2
Definition: A function of the form
p ( x)
f ( x) =
,
q( x)
where p ( x) and q ( x) are polynomial
L20
4.5 Rational Functions
*See Appendix for important graphs you need to
memorize*
Definition: A function of the form
p ( x)
f ( x) =
,
q( x)
where p ( x) and q ( x) are polynomials, is called a
rational function.
Recall:
The rational function f is unde
L19
4.1 Quadratic Functions
Example. Sketch the graph of the function
f ( x) = 1 ( x 3) 2 + 2
2
A function f is called a quadratic function if
f ( x) = ax 2 + bx + c ,
where a, b, and c are real numbers and a 0 .
Example. Sketch the graph of the function
L19
4.1 Quadratic Functions
A function f is called a quadratic function if
f ( x) = ax 2 + bx + c ,
where a, b, and c are real numbers and a 0 .
Example. Sketch the graph of the function
g ( x) = ( x + 2) 2 + 3
162
Example. Sketch the graph of the functio
L18
3.7 Operations and Compositions
Definition. Given two functions f and g. Let D f and
Dg be the domains of f and g.
Then for all x D f Dg ,
( f g )( x) = f ( x) g ( x)
( fg )( x) = f ( x) g ( x) ,
and for all x D f Dg g ( x) 0 ,
Example. Let f ( x) = x
L18
3.7 Operations and Compositions
Definition. Given two functions f and g. Let D f and
Dg be the domains of f and g.
Then for all x D f Dg ,
( f g )( x) = f ( x) g ( x)
( fg )( x) = f ( x) g ( x) ,
and for all x D f Dg g ( x) 0 ,
f
f ( x)
( x) =
.
g
g (
L17
3.5 Graphs of Relations and Functions
A piecewise-defined function is a function defined by
different rules over different parts of the domain.
Continuity:
To sketch the graph of a piecewise function:
A function is continuous over an interval of its d
L17
3.5 Graphs of Relations and Functions
Continuity:
A function is continuous over an interval of its domain
if its hand-drawn graph over that interval can be
sketched without lifting the pencil from the paper.
The graph of a continuous function has no h
L16
3.3 Linear Functions
Definition. A function f is called a linear function if
f ( x) = ax + b ,
where a and b are real numbers.
Important: The next definitions apply to any relation
F ( x, y ) = 0
x-intercept(s): The x-value(s) for which the graph
inte
L16
3.3 Linear Functions
Definition. A function f is called a linear function if
f ( x) = ax + b ,
where a and b are real numbers.
Important: The next definitions apply to any relation
F ( x, y ) = 0
x-intercept(s): The x-value(s) for which the graph
inte
L15
3.2 Functions
Definition. A function is a relation such that for each
element in the domain there is only one element in the
range.
Example. Determine which of the following relations
are functions?
b) y = x
a) y = x 2
Function Notation: y = f ( x ) .
L15
3.2 Functions
Definition. A function is a relation such that for each
element in the domain there is only one element in the
range.
Function Notation: y = f ( x ) .
If x is an element in the domain of f , then y = f ( x ) is
the corresponding element
L14
9.1, 9.5 Linear and Nonlinear
Systems of Equations
Definition: A linear system of equations is a set of
equations, in which all the variables are at most raised
to the first power. None of the variables are multiplied
together.
Example. Solve the line
L14
9.1, 9.5 Linear and Nonlinear
Systems of Equations
Definition: A linear system of equations is a set of
equations, in which all the variables are at most raised
to the first power. None of the variables are multiplied
together.
The solution set of a s
L13
3.1 Relations and the Rectangular
Coordinate System; Circles
Example. Find the domain and range of each relation.
a) y = 2 x
An ordered pair is given as ( x, y ) .
Definition. A set of ordered pairs is called a relation.
The domain of a relation is th
L13
3.1 Relations and the Rectangular
Coordinate System; Circles
An ordered pair is given as ( x, y ) .
Definition. A set of ordered pairs is called a relation.
The domain of a relation is the set of all first elements
in the ordered pairs cfw_ x . The ra
L12
2.8 Absolute Value Equations &
Inequalities
Example. Solve for x :
a) 1 x = 3
Recall:
1. x is the distance on a number line from x to 0.
2. x > 0 if x 0 , and x = 0 if and only if x = 0 .
3. x = x .
x if x 0
.
4. The algebraic definition: x =
x if
Appendix
*Basic Functions and Relations*
I. f ( x ) = x
Domain: (, )
Range: (, )
Increasing:
(, )
Decreasing:
None
Continuity:
(, )
II. f ( x) = x 2
Domain: (, )
Range: [0, )
Increasing:
(0, )
Decreasing:
(,0)
Continuity:
(, )
I
Appendix
III. f ( x) = x3