determine the vertical asymptotes of the graph of the function. give your answers as equations in exact…

determine the vertical asymptotes of the graph of the function. give your answers as equations in exact form.\n\n( h ( a ) = \frac { a + 1 } { 3 a ^ { 2 } - 7 a + 2 } )\n\nseparate multiple equations with commas as necessary. select \none\ if applicable.\n\nequation(s) of the vertical asymptote(s):

determine the vertical asymptotes of the graph of the function. give your answers as equations in exact form.\n\n( h ( a ) = \frac { a + 1 } { 3 a ^ { 2 } - 7 a + 2 } )\n\nseparate multiple equations with commas as necessary. select \none\ if applicable.\n\nequation(s) of the vertical asymptote(s):

Answer

Explanation:

Step1: Factor the denominator

Factor (3a^{2}-7a + 2). Using the formula (ax^{2}+bx + c=a(x - x_1)(x - x_2)) where (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (3a^{2}-7a + 2) ((a = 3), (b=-7), (c = 2)), we have (a=\frac{7\pm\sqrt{49-24}}{6}=\frac{7\pm5}{6}). So (3a^{2}-7a + 2=(3a - 1)(a - 2)).

Step2: Find the vertical asymptotes

The vertical asymptotes of a rational function (y=\frac{f(a)}{g(a)}) occur at the values of (a) that make (g(a)=0) (provided (f(a)\neq0) at those values). Set (3a - 1=0) gives (a=\frac{1}{3}), and set (a - 2=0) gives (a = 2). Also, check (h(\frac{1}{3})=\frac{\frac{1}{3}+1}{0}) (undefined) and (h(2)=\frac{2 + 1}{0}) (undefined) and (a+1\neq0) when (a=\frac{1}{3}) or (a = 2).

Answer:

(a=\frac{1}{3},a = 2)