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: 16
Grade code: 1.1.2.LI.1
Strand code: 1
Sub-strand code: 2
Content standard code: 1.1.2.CS.1
Indicator code: 1.1.2.LI.1
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATIONS OF ALGEBRA
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This lesson introduces an important family of functions called rational functions. Think about them as "fractions of algebra". Just like we can't divide by zero with numbers, there are certain values we can't use in these functions. Understanding this helps us solve real-world problems. For example, a small business owner in Kejetia Market making plantain chips needs to calculate the average cost per bag. This calculation uses a rational function! By understanding the "forbidden" values (the domain), we can avoid impossible situations and make better business decisions. This topic builds directly on your knowledge of polynomials and solving equations.
What is a Polynomial? (Quick Revision) Before we can understand a rational function, we must remember what a polynomial is. A polynomial is an expression with variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents. Examples of polynomials: `p(x) = 3x + 7` (Linear polynomial) `q(x) = x² - 5x + 6` (Quadratic polynomial) `k(x) = 8` (Constant polynomial) Examples of expressions that are NOT polynomials: `√x + 2` (because the exponent of x is ½) `5/x` (because this is `5x⁻¹`, which has a negative exponent) Defining a Rational Function A rational function is a function that can be written as a fraction of two polynomials. If `p(x)` and `q(x)` are polynomials, then a rational function `R(x)` has the form:
``` p(x) R(x) = ------ q(x) ``` ...with one very important rule: The denominator, `q(x)`, cannot be equal to zero.
Analogy: Think of regular fractions like `1/2`, `3/4`, `10/5`. The word "rational" comes from "ratio". A rational function is simply a *ratio* of two polynomials.
Example: Is `f(x) = (2x + 1) / (x - 3)` a rational function? The numerator is `p(x) = 2x + 1`. This is a polynomial. The denominator is `q(x) = x - 3`. This is also a polynomial. Yes, this is a rational function. The Golden Rule: The Denominator Cannot Be Zero! In mathematics, division by zero is undefined. You can try it on your calculator: `10 ÷ 0` will give you a "Math Error". This is the most important concept for finding the domain of a rational function.