Determine if each function is onetoone If it is find its inv

Determine if each function is one-to-one. If it is, find its inverse; f(x)= x+4/3x-5

Solution

We have f(x) = (x+4 )/(x + 3) let y = f (x) = (x+4 )/(x + 3) . Then y(x + 3) = x + 4 ( assuming that x -3) or, xy + 3y = x + 4 or, x - xy = 3y - 4 or, x ( 1 - y) = 3y - 4 or , x = (3y -4)/ ( 1 -y) ( assuming y 1) . On switching the values of x and y, we know that f maps ( 3x - 4)/ (1-x) R to x R i.e . x R is the image of ( 3x - 4)/ (1-x) R under f.

If x = 1, (3x -4)/(1 - x)  is indeterminate. Thus 1 is not the image of any element in R under the function f. Therefore, f is not one-to-one. Since only a one to one function can have an inverse, f does not have any inverse function.