1.3. Functions

A function is a rule assigning exactly one output in the co-domain to each input from the domain.

A function can be thought of as a machine that takes an input, processes it according to a specific rule, and produces an output. For example, if we have a function that doubles a number, inputting 2 would give us 4 as the output since 2 doubled is 4. This rule ensures that each input has only one corresponding output, making functions predictable.

Imagine a vending machine. You select a button corresponding to your choice (input), and the machine dispenses one specific item (output) for that choice. If you press the button for a soda (input), you will get that exact soda (output) every time. Just like in functions, each input (button pressed) corresponds to one unique output (item dispensed).

Explanation of the sets related to the function inputs and outputs.

The domain of a function is the complete set of possible inputs. The co-domain is a set that includes all possible outputs that could be produced by the function, while the range consists only of the actual outputs that can be produced given the domain. For instance, if our function is f(x) = x², the domain can be all real numbers, the co-domain can be all real numbers, but the range would only be all non-negative real numbers (since squaring a number can't result in a negative).

Think of a classroom where all students (domain) can answer questions. The set of all students includes everyone present. However, if we only consider the students who correctly answer questions (range), that would be a smaller group. The teacher's grading criteria (co-domain) allows for students to answer questions right or wrong, but only the correct answers are counted when determining who passed.

Different kinds of functions based on how inputs and outputs correspond.

Functions can be categorized based on how they map inputs to outputs. For example, an injective function (one-to-one) means that no two different inputs produce the same output. A surjective function (onto) covers every element in the co-domain. A bijective function is both injective and surjective, meaning every input has a unique output and all outputs are used.

Consider a student-schedule mapping: if each student has a unique class schedule with no overlaps (injective), but the schedule covers all available classes (surjective), then all students are perfectly matched to classes with no duplicates. If both conditions are true, this means students and classes are in a bijective relationship.

How two functions can be combined to form a new function.

The composition of functions involves combining two functions where the output of the first function becomes the input of the second. This is denoted as (f ∘ g)(x) = f(g(x)). This means you calculate g(x) first and then apply f to that result. For example, if f(x) = x + 1 and g(x) = 2x, then (f ∘ g)(x) would equal (2x + 1).

Imagine making a sandwich. First, you spread peanut butter (g), then add jelly (f) to create a delicious peanut butter and jelly sandwich (f(g)). Here, the jelly process is only done after the bread has been prepared with peanut butter. Each step builds on the last, just like functions can build on each other when composed.

Functions that reverse the mapping of a bijective function.

An inverse function essentially reverses the roles of inputs and outputs of a function. If f(x) takes an input and produces an output, then the inverse function f⁻¹(x) takes that output and produces the original input. Inverse functions exist if the original function is bijective. For instance, if f(x) = 2x + 3, the inverse function would involve solving for x, leading to f⁻¹(x) = (x - 3)/2.

Think of a locked box where you input a key (function) to get an item inside (output). If you want to get the key back in its original position, you'd essentially need to follow the steps backward (inverse function). The box represents the relationship where retrieving the item is the function, and putting back the key is the inverse function.

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