Invertible on the intervals question...

Question

Use analytical and/or graphical methods to determine the intervals on which the following function has an inverse. (Make each interval as large as possible.)

 

f(x)=1/(x-4)

 

The function is invertible on the interval(s) ____ 

 

Please explain! thanks!

Answer 1

 

The domain of function f is any real number EXCEPT 4. Therefore, the vertex asymptote is at x=4

 

To prove the function is 1-to-1 everywhere else, we have

     1/(A-4) = 1/(B-4) for elements A,B in the domain; that is NEITHER A nor B is 4

 

 cross multiplication shows that B-4 = A-4

                                              So B=A after adding 4 to both sides

  Therefore, if the function has the same y-values then the x-values MUST also be the same, 

  which proves the function is 1-to-1

 

So the function is invertible everywhere except at the discontinuity x=4

 

The inverse is calculated as follows:

 y = 1/(x-4) <--- given

 

 x = 1/(y-4) <--- swaps x and y

 

x(y-4) = 1

 

y-4 = 1/x

 

y = 1/x + 4

 

The long term behavior (as x--->infinity) is in fact y=4 , the horizontal asymptote

 

Finally to prove the inverse   f(g(x)) = (1/x + 4-4)^(-1) =  (1/x)^-1 = x

     and g(f(x)) = [(x-4)^(-1)]^(-1) + 4 = x-4 + 4 = x

where g is the stated inverse of f

 

The function is invertible on the interval (-infinity,4) and (4,infinity)