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Algebraic Functions and their properties

What do you understand by a function and what are its properties?
Yes, it’s initially a bit confusing but once you understand the concept correctly you won’t ever get confused again. The flowery definitions make it more confusing at times hence we will stick to a simpler but accurate way of defining a function. Before, we move to function, it’s better to know a little about relation. An algebraic ‘relation’ is a connection between two sets of information. The starting set of information is called ‘domain’ and the set of output is known as ‘range’. A single domain may lead to several range of outputs related through that particular ‘relation’.
But an algebraic ‘function’ slightly differs from a ‘relation’. It’s a specific and precise relation that leads to one and only one value of range for a single domain. It’s technically defined as an equation which gives only one value of ‘y’ for a single value of ‘x’. That is for any value that is represented by ‘x’ (domain) will yield only a single corresponding value of ‘y’ (range).

Thus, every function is a relation but every relation is not a function.
However, a very important fact to remember here is that the corresponding value of range i.e. ‘y’ can be same for two different values of domain i.e. ‘x’ when calculated through a particular algebraic ‘function’.
Ex:  y=x²+3, this is a function for any value of ‘x’. But, y²=x+1 i.e. y=±√(x+1), is not a function.

A function of ‘x’ can be denoted by a separate variable ‘y’ or also by f(x). This f(x) is called “function notation”. Functions can be further denoted by g(x), h(x) etc. what matters is the other side of the equation should have a mathematical operation involving ‘x’. Ex: g(x) = √(x²-9)

It’s easier to understand while plotting a graph for a particular function when ‘y’ and ‘x’ values are used as you can directly plot them on the y-axis and x-axis respectively. A function is thus described as an equation or a graph where no two pairs of values will have the same x-coordinates. This can also be called a property of a function.
In other words, if you draw a vertical line through the plotted graph of a function, it should not intersect the graph more than once; else it’s not a function. This is also referred as the vertical line test.

For a function f(x), it’s not necessary that every possible value of domain (x) will yield a range, f(x), but it’s mandatory that whenever it does it stays unique. Ex:  f(x) = 5/x, this doesn’t hold good when (x=0), hence this function would have a condition attached as f(x)=5/x when x≠0.
Then, we also have “piecewise function”, which is a function expressed into two or more pieces and which one to follow depends on the value of ‘x’.  Ex:  f(x) = (x+1) if (x>5) & (x–1) if (x≤5)

That’s a basic comprehensive idea of algebraic functions and their properties, however there’s much more to it which will be covered elsewhere.