WAEC SSCE Further Mathematics
Study notes for Polynomial Functions — part of the WAEC SSCE Further Mathematics syllabus. 8 learning objectives with explanations and exam tips.
When you add or multiply polynomials together, you always get another polynomial back—this is called closure. Think of it like mixing two Nigerian dishes; you still get food! Polynomials also follow the commutativity rule, meaning the order doesn't matter: adding or multiplying 3x + 2 and x² gives the same result whether you do it this way or that way. Associativity means grouping doesn't matter either—(2x + 3x) + 4x equals 2x + (3x + 4x).
Distributivity is when you expand brackets: multiply each term inside by the term outside. For example, 2(x² + 3x) becomes 2x² + 6x. The identity element for addition is zero (adding zero changes nothing), while the multiplicative identity is one (multiplying by one changes nothing). Every polynomial has an additive inverse—its negative version that cancels it out.
When you work with polynomial functions in Further Mathematics, you need to understand how mathematical statements connect logically. The rule of syntax means checking whether your mathematical statements follow proper rules—just like grammar in English. For instance, if you claim that "if a polynomial has degree 3, then it has exactly 3 real roots," you must verify this is actually true by testing cases.
Think of it like a Nigerian court case: you present evidence (premises) to reach a conclusion (deduction). If your polynomial f(x) = x³ - 2x has a root at x = 0, this is a true statement you can use in further arguments. Implications work similarly—if one thing proves another, you can confidently deduce the result. The key is ensuring each logical step follows from valid mathematical rules, not just assumptions.
A truth table is simply a systematic way of showing all possible input and output values for a logical statement or function. Think of it like a record book that displays every combination of true or false values and what result you get from each combination.
Imagine you're a JAMB registration officer at your school deciding who gets admitted. Your rule might be: "A student must have passed Mathematics AND English." Using a truth table, you'd list every possible scenario—student passed both subjects, passed only Math, passed only English, or failed both—then show whether they qualify for admission in each case.
In polynomial functions, truth tables help you verify whether certain values satisfy your equations. They're particularly useful when dealing with logical operators and conditional statements in your Further Mathematics curriculum.
Think of a function like a machine at a factory. The domain is all the raw materials (inputs) the machine can accept, while the co-domain is the complete set of possible outputs it's designed to produce.
For polynomial functions like f(x) = 2x + 3, the domain is usually all real numbers because you can substitute any number for x. However, the range (actual outputs) might be different from the co-domain.
Consider a tailor's pricing function in Lagos: if f(x) = 500x + 2000 where x is the number of shirts, the domain would be non-negative integers (you can't make negative shirts), but the co-domain could be all positive real numbers representing prices in naira.
The key difference: domain is what you're allowed to input, while co-domain is what the function could theoretically produce. The range is what it actually produces.
A mapping is simply a relationship between two sets where each input produces an output. Think of it like a registration process at your school where each student (input) gets assigned a class (output). Four special types exist that WAEC loves testing.
An identity mapping is when every element maps to itself, like f(x) = x. A constant mapping sends all inputs to one single output, such as f(x) = 5 where everything becomes 5. A one-to-one (injective) mapping means different inputs never produce the same output—like how each JAMB candidate gets a unique registration number. An onto (surjective) mapping ensures every element in the output set receives at least one input, similar to how every senior class in Nigeria must have at least one student.
When a function is both one-to-one and onto, it's called bijective, and these are the most important for solving real problems.
Think of a function as a machine that transforms input into output. The inverse function reverses this process, taking you back from output to input. If function f changes x into y, then the inverse function f⁻¹ changes y back into x.
For example, imagine a Nigerian trader who converts naira to dollars using a fixed rate. The function might be f(x) = 50x, where x represents naira and gives you dollars. The inverse function f⁻¹(x) = x/50 converts dollars back to naira. Both functions "undo" each other.
To find an inverse, swap x and y in your equation, then solve for the new y. Not every function has an inverse—it must be one-to-one, meaning each output comes from exactly one input.
When you combine two functions together, you create what we call a composite function. Think of it like this: if function f changes an input into an output, and then function g takes that output as its new input, you've created a composite function written as (g∘f)(x) or g[f(x)].
Imagine a student's journey to school. First, the student walks from home to the bus stop—that's function f. Then, the bus carries them from the bus stop to school—that's function g. The complete journey from home to school is the composite function.
Mathematically, if f(x) = 2x + 1 and g(x) = x², then (g∘f)(x) means you substitute the entire f(x) into g. So (g∘f)(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
The order matters greatly. (g∘f)(x) is different from (f∘g)(x), just like walking then taking a bus differs from taking a bus then walking.
A linear function is simply a mathematical relationship between two variables that forms a straight line when you draw it on a graph. Think of it like calculating your transport fare in Lagos: if okada costs ₦100 per kilometre plus ₦50 base fare, your total cost is expressed as f(x) = 100x + 50, where x is distance in kilometres. This is a linear function because it follows the pattern f(x) = mx + c, where m is the slope and c is the y-intercept.
Linear equations are solved to find exact values, like finding how many kilometres you can travel with ₦1,000. Linear inequalities, however, show a range of solutions using symbols like < or >. For example, if you want to spend less than ₦500, you'd write 100x + 50 < 500 and solve to get x < 4.5 kilometres. Understanding the difference between equations and inequalities is crucial for WAEC.