Question

When will be f(x) = g(y), if f(x) = g(y) does it mean x=y?

Answer 1

Task. If $f(x)=f(y)$ does it mean $x=y$?

Solution. Thus we have a function $f:A\to B$ between two sets $A$ and $B$. Also we have two points $x,y\in A$ such that $f(x)=f(y)$. The question is whether $x=y$.

The answer is that “this depends on $f$”. It is possible to construct examples when $f(x)=f(y)$ implies $x=y$, and examples when $f(x)=f(y)$ does not imply $x=y$.

Example 1: $f(x)=f(y)$ implies $x=y$.

Define a function $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=2x$. Then if $f(x)=f(y)$ for some $x,y$, that is

$2x=2y$

then

$x=y.$

Example 2: $f(x)=f(y)$ does not imply $x=y$.

Define a function $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=x^{2}$. Notice that

$f(1)=f(-1)=1^{2}=(-1)^{2}=1$

however

$1\neq-1.$