Which graph represents the function f(x)= 2/x-1 +4

1) The expression needs some parenthesis for fully understanding.

I will rewrite it in a way that does not permit doubts:

[tex]f(x)= \frac{2}{x-1} + 4[/tex]


2) It might help  you to know that it is the same that g(x-1)+4 being g(x) = 2 / x.

That helps becuase it means that the graph of f(x) is the graph of the function 2 / x shifted 1 unit to the right and 4 units upward.

3) Also, this will help to determine the graph of f(x):

- y-intercept: =>. x = 0 => f(0) = -2 + 4 = 2

- x - intercept (roots)  => y = 0 => 0 = 2 / (x - 1) + 4 => x = 1/2

- domain: all real values - {1}

- vertical asymptotes: find the limit for when x approaches to 1. Given that it is infinite x = 1 is a vertical asymptote

- ending behavior: find the limits when x → +/- ∞.

It is 4, so y = 4 is horizontal asymptote for both extremes: f(x) appraches 4 for the two extremes without touching the line y = 4

- range all real values - {4}

- it does not have regional minima o maxima.

4) Finally do a table with some values.

I have attached a graph of the function, which you can understand and do by yourself with the information and explanations given above.