What is a ‘Function’ in Mathematics? What is it that we mean when we write ‘function of x’ as f(x)?

Suppose a teacher is teaching a subject to five students. Even though the subject is the same, each person will value it differently according to the way they think, in other words, according to the way their brain functions. In the same way, if we take the subject as ‘x’ and it goes into five students – a, b, c, d and e, the way each of them function will be different because their brain functions differently. The way ‘a’ functions can be called as a(x), for ‘b’ we can call b(x) and so on.
Imagine a stock broker, Mr. f, who tells his customer, “Whatever money you give me to invest, I’ll return you Rs.100 less than thrice the amount”. For this statement, mathematically we write: f(x) = 3x – 100, where x is ‘whatever money is given’. Another stock broker, Mr. g, tells his customer, “I’ll return you double the money what you give me plus Rs.100 more.” This function can be written as g(x) = 2x + 100. Now, assume that you are the customer. You start thinking, “Who can give me more money?” If you give Rs.100 to Mr. f, he will give you according to his function, i.e., 3x – 100 = 3 x 100 – 100 = Rs.200. If you give the same amount of Rs. 100 to Mr. g, he will give you 2 x 100 + 100 = Rs. 300. So, depending on the different functions, the results differ for the same input ‘x’, in this case Rs.100.

Function of x means, how does x act in a function with the name f. This is what f(x) means. g(x) means, how x acts in another function with the name g. Basically, the mathematical word function has the meaning of the ordinary word function – ‘the way something works’.

To represent different functions (as we saw in this case of representation of different people functioning differently), we use different letters and write as f(x), g(x), h(x), etc.