What is the composition of functions?

The composition of functions is the process of applying one function to the results of another function.

In more detail, the composition of functions involves taking two functions, say \( f \) and \( g \), and creating a new function by applying one function to the output of the other. This is often written as \( (f \circ g)(x) \), which means \( f(g(x)) \). Here, you first apply the function \( g \) to the input \( x \), and then apply the function \( f \) to the result of \( g(x) \).

For example, if \( f(x) = 2x + 3 \) and \( g(x) = x^2 \), the composition \( (f \circ g)(x) \) would be \( f(g(x)) \). First, you find \( g(x) \), which is \( x^2 \). Then, you apply \( f \) to this result: \( f(x^2) = 2(x^2) + 3 = 2x^2 + 3 \). So, \( (f \circ g)(x) = 2x^2 + 3 \).

It's important to note that the order in which you compose the functions matters. \( (f \circ g)(x) \) is generally not the same as \( (g \circ f)(x) \). Using the same functions as before, \( (g \circ f)(x) \) would be \( g(f(x)) \). First, you find \( f(x) \), which is \( 2x + 3 \). Then, you apply \( g \) to this result: \( g(2x + 3) = (2x + 3)^2 \). So, \( (g \circ f)(x) = (2x + 3)^2 \).

Understanding the composition of functions is useful in many areas of mathematics, including solving complex equations and analysing real-world problems where multiple processes occur in sequence.