for ( f(x)=\frac{5x - 3}{x^{2}-5x - 5}), (a) identify the horizontal asymptotes (if any). (b) if the graph…

for ( f(x)=\frac{5x - 3}{x^{2}-5x - 5}), (a) identify the horizontal asymptotes (if any). (b) if the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s). separate multiple equations of asymptotes with commas as necessary. select \none\ if applicable. the graph has no horizontal asymptotes. the graph has at least one horizontal asymptote. equation(s) of the horizontal asymptote(s): crossover point(s):

for ( f(x)=\frac{5x - 3}{x^{2}-5x - 5}), (a) identify the horizontal asymptotes (if any). (b) if the graph of the function has a horizontal asymptote, determine the point (if any) where the graph crosses the horizontal asymptote(s). separate multiple equations of asymptotes with commas as necessary. select \none\ if applicable. the graph has no horizontal asymptotes. the graph has at least one horizontal asymptote. equation(s) of the horizontal asymptote(s): crossover point(s):

Answer

Explanation:

Step1: Determine degree of numerator and denominator

The degree of the numerator $n = 1$ (for $5x - 3$) and degree of the denominator $m=2$ (for $x^{2}-5x - 5$). Since $n<m$, the horizontal - asymptote is $y = 0$.

Step2: Find crossover points

Set $f(x)=0$. So, $\frac{5x - 3}{x^{2}-5x - 5}=0$. A rational function is zero when its numerator is zero and its denominator is non - zero. Set the numerator equal to zero: $5x-3 = 0$. Solving for $x$, we get $x=\frac{3}{5}$. When $x = \frac{3}{5}$, the denominator $(\frac{3}{5})^{2}-5\times\frac{3}{5}-5=\frac{9}{25}-3 - 5=\frac{9}{25}-8=\frac{9 - 200}{25}\neq0$.

Answer:

Equation(s) of the horizontal asymptote(s): $y = 0$ Crossover point(s): $(\frac{3}{5},0)$