Show that the function in part (a) can be written as f(x +vt) + g(x - vt).
To show that a function can be written in the form f(x + vt) + g(x - vt), let's first consider the function f(x) and g(x).
(a) We are given a function, but we don't have its actual expression. However, we know that we can write it as f(x + vt) + g(x - vt).
Now, let's analyze the terms present in the expression f(x + vt) + g(x - vt):
1. The term (x + vt) represents a rightward shift of x units due to the positive vt term. It indicates that as time progresses (t increases), the function is moving rightwards. This can be seen as a wave-like propagation.
2. The term (x - vt) represents a leftward shift of x units due to the negative vt term. It indicates that as time progresses (t increases), the function is moving leftwards. This can also be seen as a wave-like propagation, but in the opposite direction.
By having both terms (x + vt) and (x - vt), we can capture both the rightward and leftward motion of the function, thus covering all possible positions as time evolves.
Therefore, we can conclude that the function in part (a) can indeed be written as f(x + vt) + g(x - vt), where f(x) and g(x) are suitable functions that account for the rightward and leftward motions, respectively.