Find the domain of the function and identify any vertical and horizontal asymptotes. $f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$
A)Domain: all real numbers x
Vertical asymptote: No vertical asymptoteHorizontal asymptote: y = 5
B)Domain: all real numbers except x = 0
Vertical asymptote: x = 0
Horizontal asymptote: y = -8
C)Domain: all real numbers x
Vertical asymptote: x = -5
Horizontal asymptote: y = 0
D)Domain: all real numbers except x = 5
Vertical asymptote: No vertical asymptote
Horizontal asymptote: No horizontal asymptote
E)Domain: all real numbers except x = 9
Vertical asymptote: x = 5
Horizontal asymptote: y = 8

Question

Find the domain of the function and identify any vertical and horizontal asymptotes.   $f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$  
A)Domain: all real numbers x
Vertical asymptote: No vertical asymptoteHorizontal asymptote: y = 5
B)Domain: all real numbers except x = 0
Vertical asymptote: x = 0
Horizontal asymptote: y = -8
C)Domain: all real numbers x
Vertical asymptote: x = -5
Horizontal asymptote: y = 0
D)Domain: all real numbers except x = 5
Vertical asymptote: No vertical asymptote
Horizontal asymptote: No horizontal asymptote
E)Domain: all real numbers except x = 9
Vertical asymptote: x = 5
Horizontal asymptote: y = 8

Quizplus · Accepted Answer

The denominator $x^2 + x + 9$ has no real roots, so the function is defined for all real numbers. The degree of the numerator and denominator are the same, so the horizontal asymptote is the ratio of the leading coefficients, $y = \frac{5}{1} = 5$. There are no values of $x$ that make the denominator zero, so there are no vertical asymptotes.

Find the Domain of the Function and Identify Any Vertical f(x)=5x2+8x2+x+9f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }f(x)=x2+x+95x2+8​

Find the Domain of the Function and Identify Any Vertical $f ( x ) = \frac { 5 x ^ { 2 } + 8 } { x ^ { 2 } + x + 9 }$