edited by
9,575 views
41 votes
41 votes

Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, x-y)$. The inverse function of $f$ is given by

  1. $f^{-1} (x,y) = \left( \frac {1}{x+y}, \frac{1}{x-y}\right)$

  2. $f^{ -1} (x,y) = (x-y , x+y)$

  3. $f^{-1} (x,y) = \left( \frac {x+y}{2}, \frac{x-y}{2}\right)$

  4. $f^{-1}(x,y)=\left [ 2\left(x-y\right),2\left(x+y\right) \right ]$
edited by

9 Answers

Best answer
43 votes
43 votes
to find inverse of the function take

$z_1=x+y \text{          } \to(1)$

$z_2=x-y \text{          } \to (2)$

Adding (1) and (2) we get,

$x = \frac{z_1+z_2}{2}$ and $y = \frac{z_1-z_2}{2}$

So, $f \left(\frac{z_1}{2},\frac{z_2}{2}\right) = \left(\frac{z_1+z_2}{2},\frac{z_1-z_2} {2}\right) = (x, y) \\ \implies f^{-1}(x, y) = \left(\frac{z_1}{2},\frac{z_2}{2}\right) \\=  \left\{\frac{x+y}{2},\frac{x-y}{2} \right\}$

Correct Answer: $C$
edited by
88 votes
88 votes
Taking an example:

$f(2,3)=(2+3,2-3)=(5,-1)$

$f^{-1}(5,-1)$ should be $(2,3).$

Substituting the values we get (C) as answer.
11 votes
11 votes

Answer : C

f(x,y) = ( x+y , x−y ) . for invertible function you have to find that there should be bijection (one to one correspondence) . 

 if f(a) = b then  a = $f^{-1}$(b)

apply this concept here , f(x,y) = f(x+y , x-y) , so (x,y) = $f^{-1}$(x+y , x-y) -----------(1)

lets assume

p1 = x+y ------------(i)

p2 = x-y ------------(ii)

By Adding           (i)+(ii)                    $\frac{(p1+ p2) }{2}$ = x   

By Subtracting    (i)-(ii)                    $\frac{(p1- p2) }{2}$ = y

put value in--(1) 

$\left ( \frac{( p1+p2 )}{2} , \frac{( p1-p2 )}{2} \right )$ = $f^{-1}$( p1 , p2 ) 

$f^{-1}\left ( x,y \right )$ =  $\left ( \frac{( x+y )}{2} , \frac{( x-y )}{2} \right )$

 
edited by
Answer:

Related questions

34 votes
34 votes
5 answers
1
22 votes
22 votes
5 answers
3
Kathleen asked Oct 9, 2014
13,346 views
Let $X = \{2, 3, 6, 12, 24\}$, Let $\leq$ be the partial order defined by $X \leq Y$ if $x$ divides $y$. Number of edges in the Hasse diagram of $(X, \leq)$ is$3$$4$$9$No...