Functions

When we perform operations like addition or multiplication, certain rules always apply. Closure means when you combine two numbers using an operation, you get another number in the same set. For example, adding two whole numbers always gives a whole number. Commutativity means order doesn't matter: 3 + 5 equals 5 + 3. Associativity means grouping doesn't change the result: (2 + 3) + 4 equals 2 + (3 + 4). Think of it like sharing money with friends—whether you add person A's share first or person B's doesn't change the total.

Identity elements are special numbers that don't change others when used. Zero is the additive identity because any number plus zero stays unchanged. One is the multiplicative identity. Inverses are numbers that "undo" operations. Subtracting 5 undoes adding 5. These properties form the foundation of algebra and appear constantly in SSCE questions.

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Think of a function like a machine that takes input and produces output following strict rules. In logic, we use functions to determine whether statements are true or false. When you apply the rule of syntax, you're checking if statements follow proper logical structure, like checking if a sentence makes grammatical sense.

Consider this Nigerian example: if we say "All Lagos students study hard, Chioma is a Lagos student, therefore Chioma studies hard." This deduction is valid because it follows the rule of logic. The implication connects our premises to reach a true conclusion.

Understanding implications means recognizing that if statement A is true and leads to statement B, then B must also be true. This chain of reasoning helps you solve complex function problems correctly.

A truth table is a systematic way to show all possible outcomes of a logical statement. Think of it like a record book that documents every possibility when you combine simple true or false statements. Each row represents a different combination, and you check whether the final result is true or false.

Consider this Nigerian example: "If it rains AND the power supply is stable, then I can watch the WAEC results online." You'd create a table showing four scenarios: both conditions true, rain true but power false, rain false but power true, and both false. Only when both conditions are true is the outcome true.

Truth tables become essential when dealing with logical operators like AND, OR, and NOT. They help you visualize complex logical relationships clearly before drawing conclusions. This foundation prepares you for more advanced topics in logic and Boolean algebra.

Think of a function as a machine that takes inputs and produces outputs. The domain is simply all the input values your function can accept, while the co-domain is the set of all possible output values you're allowed to produce.

Consider a taxi fare calculator in Lagos: if the domain is distances from 0 to 50 kilometres, these are all the distances a passenger might travel. The co-domain could be all amounts from ₦0 to ₦5,000, representing possible fares the meter can display. However, not every amount in the co-domain needs to be used—this is crucial. The range (actual outputs produced) might only be certain values like ₦250, ₦500, ₦750, and so on.

Understanding this distinction helps you identify which values are valid inputs and which outputs are theoretically possible. Many WAEC questions test whether you can spot invalid domain values or understand why a co-domain differs from a range.

A function is like a machine that takes input and produces output. When we talk about special types of functions, we need to understand four key types.

A one-to-one function means each input gives a different output—no two inputs produce the same result. Think of it like assigning phone numbers to students; each student gets exactly one unique number.

An onto function (also called surjective) means every possible output value is used. If your teacher gives marks from 0-100 and every mark in that range is given to at least one student, that's onto.

An identity function is simple: whatever goes in comes out unchanged, like f(x) = x. A constant function always gives the same output regardless of input, like f(x) = 5.

A function that is both one-to-one and onto is called bijective—it's a perfect matching between inputs and outputs.

Think of an inverse function as an "undo" operation. If a function f takes you from point A to point B, then the inverse function f⁻¹ takes you back from B to A. For example, if a function converts Nigerian naira to US dollars, the inverse function converts dollars back to naira.

To find an inverse function, you swap x and y in the original equation, then solve for y. Consider f(x) = 2x + 3. Replace y with f(x), so y = 2x + 3. Swap to get x = 2y + 3, then solve: y = (x - 3)/2. Therefore f⁻¹(x) = (x - 3)/2.

Remember that only one-to-one functions have inverses—each input must produce exactly one output. When you compose a function with its inverse, f(f⁻¹(x)) = x always.

When you have two functions, you can combine them by feeding the output of one function into another. This is called a composite function. If you have functions f and g, the composite function f∘g means you apply g first, then apply f to that result. Think of it like a process: a student writes an exam (function g), then the teacher marks it and assigns a grade (function f). The final grade depends on both processes working together.

For example, if f(x) = 2x + 1 and g(x) = x², then f∘g(x) means you find g(x) = x² first, then apply f to get f(x²) = 2(x²) + 1 = 2x² + 1. Remember that f∘g is different from g∘f; order matters significantly.

A linear function is simply a mathematical relationship where one quantity depends on another in a straight-line pattern. Think of it like this: if you're selling pure water in Lagos and earn ₦50 per sachet, your total earnings depend directly on how many sachets you sell. If you sell x sachets, you earn 50x naira. That's a linear function written as f(x) = 50x.

Linear equations are statements showing when two things are equal, like 2x + 3 = 11. You solve these by doing the same operation on both sides to find x's value. Linear inequalities work similarly, except instead of an equals sign, you use symbols like > (greater than) or < (less than). When multiplying or dividing an inequality by negative numbers, always flip the inequality sign.