Let f:(-1, 1]=> Y:f(x) = 12, x +-2 ...
Question
Fullscreen

Let f:(-1, 1]=> Y:f(x) = 12, x +-2 and Y = range (f ). Show that f is invertible and find f-1. (x + 2)

11th - 12th Class
Maths
Solution
131
Rating
4.0 (1 ratings)
Fullscreen
SOLUTION We have [ begin{aligned} fleft(x_{1}right)=fleft(x_{2}right) & Rightarrow frac{x_{1}}{x_{1}+2}=frac{x_{2}}{x_{2}+2} & Rightarrow x_{1} x_{2}+2 x_{1}=x_{1} x_{2}+2 x_{2} & Rightarrow 2left(x_{1}-x_{2}right)=0 & Rightarrow x_{1}-x_{2}=0 & Rightarrow x_{1}=x_{2} end{aligned} ] ( therefore quad f ) is one-one. since range ( (f)=Y, ) so ( f ) is onto. Thus, ( f ) is one-one onto and therefore invertible. Let ( y in Y ). Then, there exists ( x in[-1,1] ) such that ( f(x)=y . ) [ begin{aligned} text { Now, } y=f(x) & Rightarrow y=frac{x}{(x+2)} & Rightarrow x=frac{2 y}{(1-y)} & Rightarrow f^{-1}(y)=frac{2 y}{(1-y)} end{aligned} ] Thus, we define: [ f^{-1}:[-1,1] rightarrow Y: f^{-1}(y)=frac{2 y}{(1-y)}, y eq 1 ]
Quick and Stepwise Solutions Just click and Send Download App OVER 20 LAKH QUESTIONS ANSWERED Download App for Free