Andy C. answered • 06/28/18

Math/Physics Tutor

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)