Determine which of the rational functions given below has the following feature(s).
-$x$-intercepts: $( - 4,0 )$ and $( 1,0 ) , y$-intercept: $\left( 0 , - \frac { 4 } { 3 } \right)$, vertical asymptotes: $x = - 1$ and $x = - 3$ horizontal asymptote: $\mathrm { y } = 1$
A) $f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x - 4 ) ( x + 1 ) }$

B) $f ( x ) = \frac { ( x + 1 ) ( x + 3 ) } { ( x + 4 ) ( x - 1 ) }$

C) $f ( x ) = \frac { ( x + 4 ) ( x - 1 ) } { ( x + 1 ) ( x + 3 ) }$

D) $f ( x ) = \frac { ( x - 4 ) ( x + 1 ) } { ( x - 1 ) ( x - 3 ) }$

Question

Determine which of the rational functions given below has the following feature(s).
-$x$-intercepts: $( - 4,0 )$ and $( 1,0 ) , y$-intercept: $\left( 0 , - \frac { 4 } { 3 } \right)$, vertical asymptotes: $x = - 1$ and $x = - 3$ horizontal asymptote: $\mathrm { y } = 1$
A) $f ( x ) = \frac { ( x - 1 ) ( x - 3 ) } { ( x - 4 ) ( x + 1 ) }$

B) $f ( x ) = \frac { ( x + 1 ) ( x + 3 ) } { ( x + 4 ) ( x - 1 ) }$

C) $f ( x ) = \frac { ( x + 4 ) ( x - 1 ) } { ( x + 1 ) ( x + 3 ) }$

D) $f ( x ) = \frac { ( x - 4 ) ( x + 1 ) } { ( x - 1 ) ( x - 3 ) }$

Quizplus · Accepted Answer

The correct rational function must have zeros at $x = -4$ and $x = 1$, which are the $x$-intercepts, and poles (vertical asymptotes) at $x = -1$ and $x = -3$. The function $f(x) = \frac{(x + 4)(x - 1)}{(x + 1)(x + 3)}$ meets these criteria, with the numerator giving the correct $x$-intercepts and the denominator giving the correct vertical asymptotes. The horizontal asymptote $y = 1$ is also satisfied since the degrees of the numerator and denominator are the same, leading to a leading coefficient ratio of 1.

Determine Which of the Rational Functions Given Below Has the Following