Comparison Paper
In mathematics, there are many branches of functions. You have Inverse, Linear, Quadratic, Cubic, Periodic, Monotonic, etc. In this paper we will cover Reciprocal and Rational Functions. Both can be plotted on a graph, both have asymptotes, and both have discontinuities. But just because they have a few similarities doesn’t mean that they are all the same. In this paper we will compare and contrast the two functions.

A rational function has the definition of, “Any function which can be written as the ratio of two polynomial functions.” Any time that the denominator is undefined, that means the function is undefined. Since a rational function is set up in a fraction, if the same term is in the numerator and the denominator then those two will “cancel” each other out. When two terms cancel each other out, a hole will occur at the point that both terms cancelled out at. The vertical asymptote is found by the term in the denominator. If you only have (x-2) in the denominator then you will have a vertical asymptote at 2 on the x axis, going vertical. It will be 2 because in the function the vertical asymptote is automatically given a negative sign, so the opposite of whatever your term is. The horizontal is found by any number outside of the fraction.

A reciprocal function can be defined as, “A function that models inverse variations.” Reciprocal functions can be graphed. They usually start at the top or sides of a graph; go inwards towards the origin, then bend past it and leave through the same quadrant that they entered. The x-axis is the horizontal asymptote, and the y-axis is the vertical asymptote. The vertical asymptote is found the same way as the rational function asymptote. As is the horizontal asymptote.

For both the Reciprocal and Rational Functions, you find the asymptotes the same way. You use the same formula, which is y=a/-x-h+k. The vertical asymptote is the opposite value because of the negative sign in the equation. The...

...Solving Equations and Inequalities
Introduction
One of the main goals in this course is to help students develop techniques for solving a wide variety of equations and inequalities, which in turn can be used to model real-life situations. This module will focus specifically on linear equations and inequalities.
Linear Equations
An equation can be written in many different ways and still be the same equation. For example:
3x + y = 4 original equation
y =...

...
maths for businss
TABLE OF CONTENTS
PAGE
UNIT 1
Set Theory
2
UNIT 2
Special Functions
9
UNIT 3
Functions and Graphs
27
UNIT 4
Elasticity of Demand, Supply, Income
37
UNIT 5
Types of Functions
53
UNIT 6
The National Income Models and the IS-LM Models
68
UNIT 7
Non Linear Functions and Applications
84
UNIT 8
Logarithmic Functions
105
UNIT 9
Production Functions
115
UNIT 10
Linear...

...Math Study Guide
Test 1
Solve each equation for x using any method learned in class.
(1) If there is no solution, then say so. Write your answers using complex numbers if needed.
Show/Explain your work.
(A) 20-3(x+4)=8-3x
(B)
(2)Solve each equation for x using algebra.
If there is no solution, then say so. Write your answers using complex numbers if needed.
(A)
(B)
(C)
(3) Simplify each of the following as much as...

...CALIFORNIA CONTENT STANDARDS
# of
Items
%
Algebra I
45
69%
Standard Set 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
1.1 Students use properties of numbers to demonstrate whether assertions are true or false.
1/2**
2.0* Students understand and use such operations as taking the opposite, finding the reciprocal,...

...Preliminaries
Integer Exponents In this section we will start looking at exponents and their properties.
Rational Exponents We will define rational exponents in this section and extend the properties from the previous section to rational exponents.
Real Exponents This is a short acknowledgment that the exponent properties from the previous two sections will hold for any real exponent.
Radicals Here we will define radical notation and relate radicals to rational exponents. We...

...4 Vedas: Repositories of Ancient Indian Lore 2.5 A Rational Approach to Study Ancient Literature 2.6 Shanghai Rankings and Indian Universities 2.7 Conclusions derived on Vedic Mathematics and the Calculations of Guru Tirthaji - Secrets of Ancient Maths
Chapter Three
31 33 50 55 58 59 60 61
INTRODUCTION TO BASIC CONCEPTS AND A NEW FUZZY MODEL
3.1 Introduction to FCM and the Working of this Model 3.2 Definition and Illustration of Fuzzy Relational Maps (FRMS) 3.3...

...MPM2D
Unit 2: Quadratic Relations
Chapter 3.2 – Properties of Graphs of Quadratic Relations
Student Name: Date:
Instruction: Students are asked to answer the following questions on this file and submit it on moodle
along with your GSP file. Type your name and the date on the top of this page. Label this
file as “yourname_worksheet” , and your GSP file as...

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