Question

Find all the critical points of the function TexFormula1" title=" f(x)= (x+1)/(x-3) " alt=" f(x)= (x+1)/(x-3) " align="absmiddle" class="latex-formula"> and determine the intervals in which the function is increasing and in which it is decreasing?

Answer 1

The function

... y = 1/x

has derivative

... y' = -1/x²

which has no zeros. It is undefined at x=0, the only critical point. The derivative is negative for all values of x, so the function is decreasing everywhere in its domain.

Your function

... y = (x+1)/(x-3)

can be written as

... y = 1 +4/(x-3)

which is a version of y = 1/x that has been vertically scaled by a factor of 4, then shifted 1 unit up and 3 units to the right. Shifting the function to the right means x=3 is excluded from the domain (and the interval on which the function is decreasing).

The critical point is x=3.

The function is decreasing on (-∞, 3) ∪ (3, ∞), increasing nowhere.