I One dimensional wave, function of a wave

[Taylor_1989]

I am currently reading through 'Optics' by Eugene Hecht chp 2 page 20, he talks about the function of the wave and the direction of travel of the wave i.e $\psi(x)=f(x-t)$ and right at the bottom of the page he say this:

Equation (2.5) is often expressed equivalently as some function of $t - x/v)$, since,

$$f(x-vt)=F(-\frac{x-vt}{v})=F(t-\frac{x}{v})$$

What I am trying to understand is the relevance of this, because he make no mentioned to this again in the chapter nor dose he refer to it in any of the exercise at then end, and just curious to now when this type of format would be used?

edit: the equation 2.5 he is referring to is: $\psi(x)=f(x-vt)$

 

[Charles Link]

For a wave traveling to the right $\psi(x,t)=f(x-vt)$. If we have the form $\psi(x,0)=f(x)$ at $t=0$, we can find the form at any other time by taking $\psi(x,t)=\psi(x-vt,0) =f(x-vt)$. $\\$ Alternatively, and this is what he is referring to, if we have $\psi(x,t)$ as a function of time at $x=0$, which is $\psi(0,t)=g(t)$, if it is a traveling wave with velocity $v$, it obeys $\psi(x,t)=g(t-\frac{x}{v})$.

 

[Charles Link]

And if you want to see an application of this, try working this homework problem that appeared on Physics Forums about a year ago. The solution is in post 2, but you might see if you can work it yourself before looking at the solution. https://www.physicsforums.com/threads/reflection-of-em-wave.934953/ Also notice it fooled a couple of other people who tried to work the problem without using the concept of your post above.