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: 11
Grade code: 1.1.2.LI.6
Strand code: 1
Sub-strand code: 2
Content standard code: 1.1.2.CS.1
Indicator code: 1.1.2.LI.6
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATIONS OF ALGEBRA
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This lesson introduces the concept of inverse functions. In our daily lives, we often perform actions that can be "undone" or "reversed." For example, the reverse of locking a door is unlocking it. The reverse of travelling from Accra to Kumasi is travelling from Kumasi to Accra. In mathematics, an inverse function does exactly this: it "undoes" the action of the original function. Understanding inverse functions is a powerful tool in algebra. It helps us solve equations and is fundamental in many fields like cryptography (coding and decoding messages), economics (modelling price and demand), and science (converting between units like Celsius and Fahrenheit).
Recap: What is a Function? A function is like a machine. You put in an input (usually 'x'), and it gives you a single, unique output (usually 'y' or f(x)). Domain: The set of all possible input values (x-values). Range: The set of all possible output values (y-values).
For a function to have an inverse, it must be a special type called a bijective function. This simply means it is both one-to-one and onto. One-to-One (Injective): Every output comes from exactly ONE input. No two different inputs give the same output. *Example:* `f(x) = x + 5`. If the output is 8, the only possible input is 3. *Non-Example:* `f(x) = x²`. If the output is 9, the input could be 3 or -3. This is not one-to-one. Onto (Surjective): The function covers all possible output values in its codomain (the set of all possible outputs).
We will focus on functions that are one-to-one, as these are the ones for which we can find a clear inverse. What is an Inverse Function? The inverse of a function `f(x)`, written as `f⁻¹(x)` (read as "f inverse of x"), is the function that reverses the effect of `f(x)`.
If a function `f` takes an input `a` to an output `b`, so that `f(a) = b`, then its inverse `f⁻¹` will take `b` back to `a`, so that `f⁻¹(b) = a`. The domain of `f(x)` becomes the range of `f⁻¹(x)`. The range of `f(x)` becomes the domain of `f⁻¹(x)`. How to Find the Inverse of a Function Algebraically We can find the inverse of a function using a reliable four-step method.