رياضيات 3 - ثالث ثانوي - درس : الدوال

Summary

This lesson covers mathematical sets and functions, starting with the classification of number sets, including natural numbers, whole numbers, integers, rational numbers, and real numbers. It explains how elements of smaller sets belong to larger sets. The lesson demonstrates how to use set notation, intervals, and relationships between sets. It also discusses how to determine if a relation is a function, and provides examples of solving for values in functions, identifying domains, and working with multi-defined functions. The lesson concludes with practical examples and step-by-step solutions to clarify key mathematical concepts.

Takeaways

• 📚 The lesson starts by introducing the concept of number sets, beginning with natural numbers, whole numbers, integers, rational numbers, and real numbers.
• 🔢 Natural numbers are denoted by 'N' and include positive numbers starting from 1 (e.g., 1, 2, 3).
• ➕ Whole numbers, represented by 'W', include all natural numbers and zero.
• ➖ Integers are symbolized by 'Z' and include positive, negative numbers, and zero.
• 📊 Rational numbers (fractions) are represented by 'Q', consisting of numbers that can be expressed as a ratio of two integers.
• 📐 The set hierarchy implies that every number in a smaller set belongs to all the larger sets above it.
• 💡 The first example explains how to define a set using a characteristic property with the format: 'x | condition, x ∈ R'.
• 🔗 The second example focuses on interval notation, distinguishing between open and closed intervals based on inequalities.
• 🔄 Analyzing functions requires checking that each input has only one output, with no repetition in the 'x' values.
• ✏️ The lesson covers solving equations and how to determine if an equation represents a function by isolating variables and verifying the domain.

Q & A

What are the main sets of numbers mentioned in the lesson?

The lesson covers natural numbers (ℕ), whole numbers (𝕎), integers (ℤ), rational numbers (ℚ), and real numbers (ℝ).

What is the difference between natural numbers and whole numbers?

Natural numbers (ℕ) consist of positive numbers starting from 1 (1, 2, 3...), while whole numbers (𝕎) include all natural numbers and 0 (0, 1, 2, 3...).

How are rational numbers defined?

Rational numbers (ℚ) are numbers that can be expressed as a fraction of two integers, where the denominator is not zero.

What does the arrow notation between sets represent?

The arrows show that if a number belongs to a certain set, it also belongs to all larger sets in the hierarchy. For example, an integer belongs to the rational and real number sets as well.

What is an example of a number belonging to multiple sets?

The number -2 belongs to the set of integers (ℤ), rational numbers (ℚ), and real numbers (ℝ).

What is the key difference between open and closed intervals?

Open intervals are used when inequalities are strict (e.g., > or <), and closed intervals are used when there is equality (e.g., ≥ or ≤).

How do you write a set using set-builder notation?

Set-builder notation is written with a variable (e.g., x), followed by a condition (e.g., x > 8), and the set to which the variable belongs, often the real numbers (ℝ).

What conditions make a relation a function?

A relation is a function if each input (x) has exactly one output (y). If an input repeats with different outputs, it is not a function.

How can you check if a graph represents a function?

You can check by drawing a vertical line. If the line crosses the graph more than once, the graph does not represent a function.

What happens when you take the square root of both sides of an equation?

Taking the square root of both sides introduces both a positive and negative solution, as indicated by the ± symbol.