with legs 
and 
is drawn with a semicircle inscribed into the triangle. What is the smallest possible radius of the semi-circle and the largest possible radius?
such that 
.
Where 
be non-negative real numbers such that 
.![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
Y by
Hello everyone,
I am learning functional equations.
To prove the below problem one -one function,I have taken two non-negative real numbers
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
I am learning functional equations.
To prove the below problem one -one function,I have taken two non-negative real numbers
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!Attachments:
This post has been edited 3 times. Last edited by Vulch, Apr 11, 2025, 8:07 PM
distinct inputs, they cannot give the same output.
is a function. 
.
is one-one then for any choice of 
and 
such that 
,
cannot be equal to 
,
.
and 
.
,
is not a correct example to contradict injectivity of the function.
such that 
.
.
are not 
,
.
be any nonzero real number, for surjectivity we need to show that there is a 
with 
.
,
and 
.
.
such that 
.
