Suppose $X \sim N(3,4^2)$ and let $Y =(X-3)/4$. Prove that $Y \sim N(0,1^2)$ as follows: Find a formula expressing $F_{Y}$ in terms of $F_{X}$. Find a formula for $f_{Y}$. Show that your formula for $f_{Y}$ agrees with the pdf of a standard normal random variable except at most at a set of measure zero. It then follows that Y has the same distribution as a standard normal random variable.
Just a quick question, $Y$'s pdf will agree with the pdf of a standard normal variable everywhere right? If not, what is the set of measure $0$ for which the pdf of $Y$ and the pdf of the standard normal variable are not equal?
If the method of proof is to express $F_Y$ in terms of $F_X$ and then to differentiate $F_Y$ to obtain $f_Y$, then yes, your density $f_Y$ will agree everywhere with the standard normal density; the set where they differ will be empty (which of course has measure zero).