Describing a function in terms of another

[Niles]

Homework Statement

I have two functions:

$$\begin{array}{l} f(x) = \left\{ {\begin{array}{*{20}c} {\pi - x\,for\,x \in (0,\pi )} \\ {0\,for\,x \in (\pi ,2\pi )} \\ \end{array}} \right. \\ g(x) = \left\{ {\begin{array}{*{20}c} {x\,for\,x \in (0,\pi )} \\ {0\,for\,x \in (\pi ,2\pi )} \\ \end{array}} \right. \\ \end{array}$$

According to my book, then g(x) = f(-x-Pi). But when I insert, I get that g(x) = Pi-(-x-Pi) = 2Pi + x for x € (0, Pi)?

 

[steelphantom]

I think it should be g(x) = f(-x + pi). There must have been a typo in the book.

g(x) = pi - (-x + pi) = pi + x - pi = x.

 

[Architeuthis Dux]

As a scientist, it is important to understand the concept of function composition. In this case, the given functions f(x) and g(x) are defined differently for different intervals of x. In order to describe g(x) in terms of f(x), we need to consider the domain of each function.

For x € (0, Pi), both f(x) and g(x) have the same definition. However, for x € (Pi, 2Pi), f(x) is equal to 0 while g(x) is equal to x. This means that for this interval, we can write g(x) in terms of f(x) as g(x) = f(x+Pi).

Let's take a closer look at the interval x € (Pi, 2Pi). When we insert x+Pi into f(x), we get f(x+Pi) = Pi - (x+Pi) = -x. This means that for this interval, g(x) is equal to -x. Therefore, we can write g(x) in terms of f(x) as g(x) = -f(x).

In summary, we can say that g(x) = f(x+Pi) for x € (0, Pi) and g(x) = -f(x) for x € (Pi, 2Pi). This is the correct way to describe g(x) in terms of f(x). It is important to pay attention to the domain of each function when performing function composition.