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An increasing function is a type of function particularly relevant in mathematics, especially in the study of calculus and analysis. To label a function as increasing, we have a simple but strict condition; the function must satisfy a specific behavior for all the pairs of numbers within its domain.

Consider you have a function, let’s call it 'f', and this function maps real numbers to real numbers. In mathematical terms, we would write this as f: R → R. Now, if you select any two numbers within this function’s domain, say x and y, where x is strictly less than y (written as x < y), the corresponding outputs must adhere to the same inequality relationship. This means f(x) < f(y). It’s a way to guarantee that as you move along the graph of the function from left to right, the function is never going down—it either remains constant or goes up.

Why is this concept important? Knowing that a function is increasing can help us predict the behavior of the function without needing to see its graph. It also plays a key role in calculus for finding function maxima or minima, and it's essential in optimization problems.

Function composition is a cornerstone concept in mathematics, especially when dealing with transformations or sequences of processes. The composition of functions refers to the combining of two functions in such a way that the output of one function becomes the input of the next.

To compose two functions, f and g, we notate this as (g ∘ f), read as 'g composed with f'. The mechanics of this process involve first applying the function f to our input, called x, and then taking that output, f(x), and using it as the input for the function g. The resulting expression, g(f(x)), represents the composition (g ∘ f).

This concept is not just abstract theory; it has practical implications in various fields such as computer science, where functions represent a sequence of instructions, or in physics, where functions might model a chain of events. Function composition allows us to understand complex systems and processes by breaking them into smaller, more manageable parts.

The domain of real numbers, often denoted as 'R', is the set of all possible values that a real number can take. When we talk about the domain in the context of functions, we're discussing the set of all possible inputs that the function can accept.

The significance of stating that a function has a domain of real numbers is in conveying that any real number can be plugged into the function and it will produce a valid outcome. This encompasses everything from negative to positive infinity, including all fractions and irrational numbers like pi or the square root of two.

Functions with real numbers domain are especially important in the real world because they can model a vast range of phenomena with continuous variables. Temperature over time, the position of a moving object, or the growth of an investment—all these can be represented with functions that take real number inputs.

When we work with inequalities, there are certain properties that help us make logical deductions and manipulations to solve them. These properties are foundational rules of how numbers and operations work.

For example, one such property is the transitive property, which tells us that if a < b and b < c, then it must follow that a < c. Another essential property is the preservation of inequality by functions that are increasing. This means, as shown in our exercise, if you have an increasing function f, and two real numbers a and b such that a < b, then you also have f(a) < f(b).

These properties keep our mathematical reasoning consistent and allow us to handle more complex situations, such as the composition of functions, without getting lost in the math. They form the fundamental rules that all students of mathematics need to understand in order to work with inequalities and functions effectively.