Reciprocal Functions

Reciprocal of Linear and Quadratic Functions

The reciprocal of a function \(f(x)\) is given by \(g(x) = \frac{1}{f(x)}\).

Reciprocal of a Linear Function

Consider a linear function \(f(x) = mx + b\). The reciprocal is \(g(x) = \frac{1}{(mx + b)}\).

Graphing:

• If \(f(x) = 0\), then g(x) is undefined, resulting in a vertical asymptote.
• If \(f(x)\) is positive, then g(x) is also positive.
• If \(f(x)\) is negative, then g(x) is also negative.

Example

Let’s graph the reciprocal of the function \(f(x) = 2x + 1\).

• Identify the zeros of \(f(x): x = \frac{-1}{2}\) (vertical asymptote).
• Determine the sign of \(f(x)\): negative for \(x < \frac{-1}{2}\), positive for \(x > \frac{-1}{2}\).
• Sketch the graph of g(x) using this information.

Sampling and types of Sampling

1) Introduction to Sampling

1.1)            What is sampling?

Welcome to the "Sampling and Types of Sampling" course! In this chapter, we will be exploring the concept of sampling and its importance in various fields of study. Sampling is a fundamental technique used to gather data from a larger population in a systematic and efficient manner.

But what exactly is sampling? Let's break it down:

How to wright news article?

Writing News Articles

Writing a news article involves addressing the five W's and one H: Who, What, When, Where, Why, and How. These elements provide a comprehensive and structured way to convey information. Here's a breakdown of each:

1. Who:
   • Identify the key individuals or groups involved in the news story.
   • Include names, titles, and any relevant background information.
   • Answer questions like: Who is the main subject of the story? Who is affected by the events?

Applications of Algebra- WS1

Algebra Applications Worksheet

The Study Zone

Section 1: Age-Based Questions

1. Sarah is 5 years older than her sister, Emma. The sum of their ages is 29. How old is each sister?

a. Sarah's age: b. Emma's age:

2. Five years ago, the sum of Ahmed's age and Maria's age was 30. If Ahmed is now 3 times as old as Maria, find their current ages.

a. Ahmed's current age: b. Maria's current age:

Percentages- WS1TSZ

Finding Percentages

1. What is 20% of 80?
2. 25 is what percent of 50?
3. If 15% of a number is 45, what is the number?
4. 40% of what number is 120?
5. If 35% of a number is 70, what is the number?
6. What percent of 60 is 15?

Derivative Rules-1

The Ultimate Guide to Derivative Rules: Product, Quotient, and Chain Rule

Derivatives are a cornerstone of calculus, used to analyze rates of change, slopes of curves, and dynamic systems. This guide explains three essential derivative rules: the Product Rule, the Quotient Rule, and the Chain Rule. Whether you're a student, teacher, or professional revisiting calculus, this detailed guide is designed to clarify these concepts with simple and complex examples.

1. Product Rule

Definition

If u(x) and v(x) are two differentiable functions, the derivative of their product is:

\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]

Step-by-Step Explanation

Data Management-1

Types of Graphs and Charts:

• Bar Graphs: Used to compare data between different groups or track changes over time. They are most useful when there are big changes or to show how one group compares against other groups. For example, you can use a bar graph to compare the number of customers by business role.
• Line Graphs: Reveal trends or progress over time, and you can use them to show many different categories of data. For instance, you can use a line graph to show the number of sales over a period of time.