المحاضرة الأولي للصف الثاني الثانوي مقدمة الدرس الأول الدوال الحقيقية

Main ideas / lessons

Context of the lesson

• The lecture is presented as part of the second year of secondary school curriculum.
• It focuses on functions (الدوال), introduced from the ideas of relations (العلاقات).

Relationship (العلاقة)

A relation describes a connection between:

• A first set and a second set

Key points:

• Elements in the first set (the “inputs”) can be numbers, letters, etc.
• A relation exists when elements from the first set are connected to elements in the second set (e.g., element 1 pairs with one element, element 2 pairs with another).
• The lecture emphasizes that relation setup is based on matching elements across two sets.

Function (الدالة)

A function is a special kind of relation where:

• Each element of the first set is associated with exactly one element of the second set.

Rule emphasized throughout:

• If an input (x) maps to two different outputs (y values), it is not a function.
• If an input (x) maps to no output, it is not considered to satisfy function behavior (implied through examples).

Core criterion used:

• One input → one output

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Methodology / step-by-step approach to decide “function or not”

A) From a mapping table / ordered pairs

1. List all ordered pairs in the form (x → y).
2. For each x:
   • Check whether it appears with only one y.
   • If the same x maps to multiple y values → not a function.
3. If an x has no y (not mapped at all) → treated as not satisfying function behavior.

Conclusion:

• If all x values map to exactly one y only → function
• Otherwise → not a function

B) Using a verbal “numbers/sets” check (conceptual)

• Identify the first set (x) and the second set (y).
• For every element in the x-set:
  • Verify it is linked to exactly one element in the y-set.

C) From a graph (vertical line test)

Use the vertical line test:

• Draw a vertical line and observe how many times it intersects the graph.

Rule:

• If any vertical line intersects the graph more than once → not a function
• If each vertical line intersects at most once → a function

The lecture contrasts:

• Intersects once → function
• Intersects twice → not a function

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Additional examples / concepts mentioned

Parity-related example (even/odd functions)

• The lecture mentions classification such as:
  • Odd function
  • Even function
• It also includes statements like “this is not a function,” but the subtitles are noisy and do not clearly preserve the exact formulas.

Domain (المجال)

• Domain is defined as:
  • the values of x that work for the entire function/problem.
• It is connected to “x values that don’t violate any condition.”

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Polynomial function (الدالة كثيرة الحدود) and the “sexagesimal” function

Polynomial function

It is described as a function whose expression:

• does not include negative exponents
• does not include x in the denominator

The lecture also explicitly disallows:

• roots involving x (e.g., forms like ( \sqrt{x} ) are treated as non-polynomial in the explanation)
• “x under a radical/root” and similar non-polynomial forms.

Sexagesimal function

Defined (as stated in subtitles) as a function that:

• does not contain a negative exponent
• does not have x in the denominator

Note: The subtitles suggest overlap with the polynomial-function idea, and some wording appears inconsistent due to auto-generation.

Meaning of negative exponents

• Negative exponent corresponds to the reciprocal form:
  • (x^{-n} = \frac{1}{x^n})

Non-polynomial cases

The lecture lists “incorrect cases” (٤ حالات غير صحيحة) to avoid:

• negative powers
• x in the denominator
• x inside roots (as described in the subtitles)
• forms leading to fractions involving x

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Speaker / sources featured (as identifiable)

• Primary speaker: Unclear/unnamed teacher/professor (referred to as “Professor” / “Mr.” in the subtitles)
• Other speakers or external sources: none are clearly named in the provided subtitles.