Given
[tex]R(x)=\frac{13x+13}{8x+16}[/tex]
To find:
a) The domain of R(x).
b) The vertical asymptote.
c) The horizontal asymptote.
Explanation:
It is given that,
[tex]R(x)=\frac{13x+13}{8x+16}[/tex]
a) Consider
[tex]\begin{gathered} 8x+16\ne0 \\ \Rightarrow8x\ne-16 \\ \Rightarrow x\ne-\frac{16}{8} \\ \Rightarrow x\ne-2 \end{gathered}[/tex]
Hence, the domain of R(x) is,
[tex]\lbrace x|x\ne-2\rbrace[/tex]
b) To find, the vertical asymptote set the denominator equal to 0 and solve for x.
[tex]\begin{gathered} \Rightarrow8x+16=0 \\ \Rightarrow8x=-16 \\ \Rightarrow x=-\frac{16}{8} \\ \Rightarrow x=-2 \end{gathered}[/tex]
Hence, the vertical asymptote is x=-2.
c) To find the horizontal asymptote set y as the fraction of the coefficients of x in the numerator and the denominator.
[tex]\Rightarrow y=\frac{13}{8}[/tex]
Hence, the horizontal asymptote is 13/8.
Thus,
a) The domain is,
[tex]\lbrace x|x\ne-2\rbrace[/tex]
b) The vertical asymptote is x=-2.
c) The horizontal asymptote is y=13/8.
