Lesson Notes By Weeks and Term v4 - SHS 1
APPLICATIONS OF ALGEBRA
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APPLICATIONS OF ALGEBRA
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Subject: Additional Mathematics
Class: SHS 1
Term: 1st Term
Week: 10
Grade code: 1.1.2.LI.5
Strand code: 1
Sub-strand code: 2
Content standard code: 1.1.2.CS.1
Indicator code: 1.1.2.LI.5
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATIONS OF ALGEBRA
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This lesson introduces students to different types of functions, focusing on how they map elements from one set to another. We will explore the special properties that make a function "well-behaved" and useful in many real-world applications. Think about the Ghana Card system: every registered Ghanaian has a unique ID number, and no two people share the same number. This is a perfect real-life example of the kind of function we will study today. Understanding these properties (injective, surjective, bijective) is fundamental for advanced topics in mathematics, computer science, and even economics.
Recap: What is a Function? A function is a rule that assigns each input element from a set (the domain) to exactly one output element in another set (the co-domain). The set of all actual outputs is called the range. Analogy: Think of a fufu pounding machine. You put in a cassava piece (input from the domain), and the machine gives you a lump of fufu (a unique output). The machine cannot give you two different outputs for the same piece of cassava. Part 1: Types of Functions
We classify functions based on how the inputs and outputs are paired.
a) Injective Function (One-to-One) An injective function is one where every output is linked to a unique input. No two different inputs will ever produce the same output. Formal Definition: A function `f: A → B` is injective if for any two elements `a₁` and `a₂` in `A`, `f(a₁) = f(a₂)` implies that `a₁ = a₂`. Simple Terms: Different inputs always give different outputs. Ghanaian Context: The function that maps each student in your school to their unique student ID number is injective. Two different students cannot have the same ID.
How to test for injectivity: Algebraic Test: Set `f(a₁) = f(a₂)`. Solve the equation. If the only possible solution is `a₁ = a₂`, the function is injective. Graphical Test (The Horizontal Line Test): Draw the graph of the function. Draw any horizontal line (`y = c`). If no horizontal line intersects the graph more than once, the function is injective.