Question

Enter the equations of the asymptotes for the function f(x). f(x)=3x−7+2Vertical asymptote: Horizontal asymptote

Answer 1

Given

$f(x)=\frac{3}{x-7}+2$

To find the vertical asymptote and the horizontal asymptote.

Now,

The given function is,

$f(x)=\frac{3}{x-7}+2$

Taking LCM on the RHS implies,

$\begin{gathered} f(x)=\frac{3+2(x-7)}{x-7} \\ f(x)=\frac{2x-14+3}{x-7} \\ f(x)=\frac{2x-11}{x-7} \end{gathered}$

The denominator of the function f(x) is the vertical asymptote.

That is,

$\begin{gathered} x-7=0 \\ x=7 \end{gathered}$

Hence, x=7 is the vertical asymptote.

Since the degree of the numerator is equal to the degree of the denominator then, the horizontal asymptote is the ratio of the coeffcient of x in the numerator to the ratio of the coefficient of x in the denominator.

That implies,

$\begin{gathered} \text{Horizontal asymptote, y}=\frac{2}{1} \\ \text{Horizontal asymptote},\text{ y=2} \end{gathered}$

Hence, the horizontal asymptote is y=2.