1.

(i) Determine whether f(x)=(2x+3)/4 for f:RtoR is invertible or notIf so find it. (ii) Let f(x)=x^(2)+2x,xge-1 . Draw graph of f^(-1)(x) also find the number of solutions of the equation f(x)=f^(-1)(x) (iii) If y=f(x)=x^(2)-3x+2,xle1.Find the value of g'(2) where g is inverse of f

Answer»

Solution :(i) GIVEN functionsis one -one and otno, therefore it is INVERTIBLE,
`y=(2x+3)/4impliesx+(4y-3)/2:.f^(-1)(x)=(4x-3)/2`
(ii) `f(x)=f^(-1)(x)` is equivalent to `f(x)=ximpliesx^(2)+2x=ximpliesx(x+1)=0impliesx=0,-1`
HENCE two solution for `f(x)=f^(-1)(x)`

(iii) `f(x)=x^(2)-3x+2,xle1`
`FG(x)=g(x)^(2)-3g(x)+2`
`implies2=g(2)^(2)-3g(2)+2`
`impliesg(2)=0,3le1`
so `g(2)=0`
`f'(x)=2x-3`
`fg(x)=ximpliesf'(g(x)).g'(x)=1impliesg'(2)=1/(f'(g(2)))=1/(f'(0))=-1/3`


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