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: $x = 0$
domain: all real numbers
C) vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$
D) domain: all real numbers except $x = 0$ and $x = \frac { 4 } { 5 }$
vertical asymptote: $x = 0$
domain: all real numbers except $x = \frac { 4 } { 5 }$
E)
vertical asymptotes: $x = 0$ and $x = - \frac { 4 } { 5 }$.

Question

Quizplus · Accepted Answer

For the function to be defined, the denominator cannot be equal to 0. Therefore, we solve the equation: 
$$5x^2+4x=0$$
$$x(5x+4)=0$$
This gives us two critical values: $x=0$ and $x=-\frac{4}{5}$. 
Thus, the domain of the function is all real numbers except $x=0$ and $x=-\frac{4}{5}$. 
We also need to check if the function has any vertical asymptotes. If the denominator of the function approaches 0 as $x$ approaches a certain value, then there is a vertical asymptote at that value. 
As $x$ approaches 0, the denominator approaches 0 but the numerator approaches 4. Therefore, there is a vertical asymptote at $x=0$. 
As $x$ approaches $-\frac{4}{5}$, the denominator also approaches 0 but the numerator approaches $-\frac{16}{5}$. Therefore, there is a vertical asymptote at $x=-\frac{4}{5}$. 
Thus, the correct choice is B.

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

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