I don't know how to solve this.. Thanks a ton to everyone that helps

Question

State the domain using an inequality, set notation, and interval notation. For any x-value excluded from the domain, state whether the graph has a vertical asymptote or a "hole" at that x-value. Use a graphing calculator to check your answer.

 

f(x)=[x^2+2x-3]/[x^2-4x+3]

Answer 1

Ana,

      The domain of definition of a function is the set of values of x for which the assigned function exists. In this case you are given a ratio of two second degree polynomials. However, if you factor both numerator and denominator, you get

                                                                 x2 +2x-3        (x-1)(x+3)  

                                                                ------------  =  ------------- 

                                                                 x2 -4x+3         (x-1)(x-3)      

                                                                   

 

 Hence, the domain is the set of values of x for which the denominator is not zero, i.e., 

 

                                                                        (x-3)(x-1) ≠ 0 or x ≠ 3, x≠ 1

 

One notices that x=1 sets both the denominator and the numerator to 0. Therefore x=1 is a hole.  

 

x=3 is the vertical asymptote.  You can show this by proving that lim_{x-> 3}-  (x+3)/(x-3) = -∞  and lim_{x-> 3+} (x+3)/(x-3) = +∞