Question

What is the difference between the composition and the product of two functions?

Answer 1

Solution:

The difference between the composition of two functions and the product of those same two functions is significant and crucial to maintaining mathematical accuracy. When two functions are multiplied, they remain independent until their inputs have produced an output, at which point the two outputs are multiplied to become one product. This could also be simplified into [f(x)] * [g(x)], but the fact remains that they are independent of one another until the final multiplication is done outside the brackets. On the other hand, composing two functions consists of inputting an entire function into the variable of another function.

For instance, if f(x)=2x and

g(x)=x+1,

f(g(x))=2(x+1)

As such, the input goes through g(x) first, and then the resultant is used as an input into f(x).