Question

Find the equation(s) of all vertical and horizontal asymptotes for the function f left parenthesis x right parenthesis equals fraction numerator left parenthesis 4 x plus 1 right parenthesis left parenthesis 5 x minus 1 right parenthesis over denominator x squared minus 9 end fraction .

Answer 1

Answer:

The vertical asymptotes are x=3 and x=-3. The horizontal asymptote is y=20.

Step-by-step explanation:

The given function is

$f\left(x\right)=\frac{\left(4x+1\right)\left(5x-1\right)}{x^2-9}$

Equate the denominator equal to 0, to find the vertical asymptotes.

$x^2-9=0$

Add 9 on both sides.

$x^2=9$

Taking square root on both sides.

$x=\pm \sqrt{9}$

$x=\pm 3$

Therefore the vertical asymptotes are x=3 and x=-3.

The given function can be written as

$f\left(x\right)=\frac{20 x^2 + x - 1}{x^2-9}$

Degree of numerator and denominator are same.If degree of numerator and denominator are same, then horizontal asymptote is

$y=\frac{\text{Leading coefficient of numerator}}{\text{Leading coefficient of denominator}}$

$y=\frac{20}{1}$

$y=20$

Therefore the horizontal asymptote is y=20.