prove that the function f:R on Rdefined as f(x) = (x-1)(x-2)(x-3) is onto but not one - one.let a real valued function f be defined as f(x) = e to the power x + e to the power -x/e to the power x - e to the power -x.find it s inverse.

For first part, notice that f(+inf) = inf and f(-inf) = -inf

Since f is continuous, therefore all points in (-inf, +inf) have a pre-image. So function is onto.

Its not one-to-one because f(1)=f(2)=f(3) = 0, so one image has multiple preimages.

hi

A1) f:R ON R means domain R is mapped to codoamin R

it''s domain is R and range also is R  (it''s a polynomial function it''s domain and range both are R)

so range  = codomain that''s why this is onto function.

but not one to one

f(x) = 0 for x = 1 , 2 , 3   so for 3 values there is one value 0 .

A2)    f(x) = (e^2x+1)/(e^2x - 1)    by some algebraic manipulation

(e^2x - 1)y = (e^2x+1)

e^2x = (y+1)/(y-1)

take natural logarithms on both sides

x = 1/2 ln {(y+1)/(y-1)}

interchange x and y

y = 1/2 ln {(x+1)/(x-1)}   is inverse

We are  here to help you in your IIT JEE preparation.

Now you can win by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.

Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar : Click here to download the toolbar.. 

Thanks and Regards

Jitender

(askIITians expert)