Mathematical Functions - Fall 2012
Quiz 1, September 16, 2012
In the following problems you are required to show all your work and provide the necessary explanations everywhere to get full credit.
2 1
+
5
2 .
1. Evaluate 1 +
1
3
+
10 15
A
x=1
B
x=2
C
x=3
Section 3.7 Rational Functions
A rational function is a function of the form
r(x) =
P (x)
Q(x)
where P and Q are polynomials.
Rational Functions and Asymptotes
The domain of a rational function consists of all real numbers x except those for which the
den
Section 1.8 Coordinate Geometry
The Coordinate Plane
Just as points on a line can be identified with real numbers to form the coordinate line, points
in a plane can be identified with ordered pairs of numbers to form the coordinate plane or
Cartesian plan
Section 2.6 Combining Functions
Sums, Differences, Products, and Quotients
Two functions f and g can be combined to form new functions f + g, f g, f g, and f /g in a
manner similar to the way we add, subtract, multiply, and divide real numbers.
EXAMPLE: T
Section 3.3 Dividing Polynomials
Long Division of Polynomials
Dividing polynomials is much like the familiar process of dividing numbers. When we divide
38 by 7, the quotient is 5 and the remainder is 3. We write
To divide polynomials, we use long divisio
Section 2.7 One-to-One Functions and Their Inverses
One-to-One Functions
HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects
its graph more than once.
EXAMPLES:
1. Functions x, x3 , x5 , 1/x, etc. are one-to-one, si
Section 2.5 Transformations of Functions
Vertical Shifting
EXAMPLE: Use the graph of f (x) = x2 to sketch the graph of each function.
(a) g(x) = x2 + 3
(b) h(x) = x2 2
EXAMPLE: Use the graph of f (x) = x3 9x to sketch the graph of each function.
(a) g(x)
Section 10.1 Systems of Linear Equations in Two Variables
EXAMPLE: Find all solutions of the system
2x y = 5
x + 4y = 7
Solution 1(Substitution Method): We solve for x in the second equation.
x + 4y = 7
x = 7 4y
Now we substitute for x in the first equat
Section 4.1 Exponential Functions
DEFINITION: An exponential function is a function of the form
f (x) = ax
where a is a positive constant.
1 x
x
10
1 3
-3
= 103 = 1000
10
2
1
-2
= 102 = 100
10
1 1
-1
= 101 = 10
10
1 0
0
=1
10
1 1
1
1
= 10
= 0.1
10
2
Section 2.2 Graphs of Functions
DEFINITION: A function f is a rule that assigns to each element x in a set A exactly one
element, called f (x), in a set B. Its graph is the set of ordered pairs
cfw_(x, f (x) | x A
EXAMPLE:
Sketch the graphs of the followi
Section 1.7 Inequalities
Linear Inequalities
An inequality is linear if each term is constant or a multiple of the variable.
EXAMPLE: Solve the inequality 3x < 9x + 4 and sketch the solution set.
Solution: We have
3x < 9x + 4
3x 9x < 9x + 4 9x
6x < 4
4
6x
Section 6.1 Angle Measure
An angle AOB consists of two rays R1 and R2 with a common vertex O (see the Figures
below). We often interpret an angle as a rotation of the ray R1 onto R2 . In this case, R1 is
called the initial side, and R2 is called the termi
Section 6.3 Trigonometric Functions of Angles
In the preceding section we defined the trigonometric ratios for acute angles. Here we extend
the trigonometric ratios to all angles by defining the trigonometric functions of angles.
Trigonometric Functions o
Section 6.2 Trigonometry of Right Triangles
Trigonometric Ratios
Since any two right triangles with angle are similar, the trigonometric ratios are the same,
regardless of the size of the triangle; the trigonometric ratios depend only on the angle .
EXAMP
Section 6.6 The Law of Cosines
The Law of Sines cannot be used directly to solve triangles if we know two sides and the angle
between them or if we know all three sides (these are Cases 3 and 4 of the preceding section).
In these two cases, the Law of Cos
Section 6.5 The Law of Sines
To solve a triangle, we need to know certain information about its sides and angles. A triangle
is determined by three of its six parts (angles and sides) as long as at least one of these three
parts is a side.
So, the possibi
Section 3.4 Real Zeros of Polynomials
Rational Zeros of Polynomials
To help us understand the next theorem, lets consider the polynomial
P (x) = (x 2)(x 3)(x + 4) = x3 x2 14x + 24
From the factored form we see that the zeros of P are 2, 3, and 4. When the
Section 10.2 Systems of Linear Equations in Several Variables
EXAMPLE: Find all solutions of the system
x+y+z =1
y+z =1
2z = 4
Solution 1(Substitution Method): We solve for z in the last equation.
2z = 4
z=2
Now we substitute for z in the second equation
Section 1.6 Modeling with Equations
EXAMPLE: A car rental company charges $30 a day and 15c a mile for renting a car. Helen
rents a car for two days and her bill comes to $108. How many miles did she drive?
Solution: We are asked to find the number of mil
Section 8.4 Plane Curves and Parametric Equations
Suppose that x and y are both given as functions of a third variable t (called a parameter) by the
equations
x = f (t),
y = g(t)
(called parametric equations). Each value of t determines a point (x, y), wh
Section 1.10 Lines
The Slope of a Line
EXAMPLE: Find the slope of the line that passes through the points P (2, 1) and Q(8, 5).
Solution: We have
m=
y2 y1
4
2
51
= =
=
x2 x1
82
6
3
1
EXAMPLE: Find the slope of the line that passes through the points P (2,
Section 4.4 Laws of Logarithms
LAWS OF LOGARITHMS: If x and y are positive numbers, then
1. loga (xy) = loga x + loga y.
x
2. loga
= loga x loga y.
y
3. loga (xr ) = r loga x where r is any real number.
EXAMPLES:
1. Use the laws of logarithms to evaluat
Section 2.4 Average Rate of Change of a Function
Suppose you take a car trip and record the distance that you travel every few minutes. The
distance s you have traveled is a function of the time t:
s(t) = total distance traveled at time t
We graph the fun
Section 4.3 Logarithmic Functions
DEFINITION: Let a be a positive number with a 6= 1. The logarithmic function with base
a, denoted by loga , is defined by
loga x = y
ay = x
So, loga x is the exponent to which the base a must be raised to give x.
4
4
4
y
Mathematical Functions - Fall 2012
Quiz 6 November 4, 2012
This part consists of 4 multiple choice problems. Nothing more than the answer is required; consequently no partial credit will be awarded.
1
1. Evaluate log5 .
7
5
A
1/7
B
1/7 Right Answer
C
7
D
Mathematical Functions - Fall 2012
Quiz 5 October 14, 2012
This part consists of 4 multiple choice problems. Nothing more than the answer is required; consequently no partial credit will be awarded.
1. Determine the end behavior of the polynomial P (x) =
Mathematical Functions - Fall 2012
Quiz 5 October 14, 2012
This part consists of 4 multiple choice problems. Nothing more than the answer is required; consequently no partial credit will be awarded.
1. Determine the end behavior of the polynomial P (x) =
Mathematical Functions - Fall 2012
Quiz 4 October 7, 2012
This part consists of 4 multiple choice problems. Nothing more than the answer is required; consequently no partial credit will be awarded.
1. For the function f (x) = x2 + 1 nd the average rate of