Given $f ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }$. Determine the domain of $f ( x )$ and find any vertical asymptotes.
A) domain: all real numbers except $x = - \frac { 4 } { 5 }$ vertical asymptote: $x = 0$
B) domain: all real numbers except $x = 0$ and $x = - \frac { 4 } { 5 }$
vertical asymptote: $x = 0$
C) domain: all real numbers
vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$
D) domain: all real numbers except $x = 0$ and $x = \frac { 4 } { 5 }$ vertical asymptote: $x = 0$
E) domain: all real numbers except $x = \frac { 4 } { 5 }$ vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$

Question

Quizplus · Accepted Answer

The domain of $f(x)$ excludes values that make the denominator zero. Factoring the denominator gives $5x^2 + 4x = x(5x + 4)$. Setting this equal to zero, we find $x = 0$ and $x = -\frac{4}{5}$ make the denominator zero, so these values are excluded from the domain. The vertical asymptote occurs where the denominator is zero but the numerator is not simultaneously zero, which is at $x = 0$.

Given f(x)=5x+45x2+4xf ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }f(x)=5x2+4x5x+4​

Given $f ( x ) = \frac { 5 x + 4 } { 5 x ^ { 2 } + 4 x }$