Question

What are the equations of the asymptotes of the graph of the function f(x)=3x^2-2x-1/x^2+3x-10?

Answer 1

The equations of the vertical asymptotes are x = -5 and x = 2. The horizontal asymptote is y = 3.

Step-by-step explanation:

The function f(x) = (3x² - 2x - 1) / (x² + 3x - 10) can be rewritten as f(x) = (3x² - 2x - 1) / (x + 5)(x - 2). To find the equations of the asymptotes, we can analyze the behavior of the function as x approaches positive and negative infinity. The vertical asymptotes occur when the function is undefined, which happens when the denominator equals zero. Therefore, the equations of the vertical asymptotes are x = -5 and x = 2. As for the horizontal asymptote, it can be found by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator. In this case, the degrees are equal, so the horizontal asymptote is y = 3/1 or y = 3.

Answer 2

An asymptote is of a graph of a function is a line that continually approaches a given curve but does not meet it at any finite distance.

There are three major types of asymptote: Vertical, Horizontal and Oblique asymptotes.

Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. They are the values of x for which a rational function is not defined.

Thus given the rational function:

`f(x)= (3x^2-2x-1)/(x^2+3x-10)`
The vertical asymptotes are the vertical lines corresponding to the values of x for which

`x^2+3x-10=0`

Solving the above quadratic equation we have:

`x^2+3x-10=0 \\ \\ (x-2)(x+5)=0 \\ \\ x-2=0 \ or \ x+5=0 \\ \\ x=2 \ or \ x=-5`

Therefore, the vertical asymptotes of the function

`f(x)= (3x^2-2x-1)/(x^2+3x-10)`
are x = 2 and x = -5


The horizontal asymptote of a rational function describes the behaviour of the function as x gets very big.
The horizontal asymptote is usually obtained by finding the limit of the rational function as x tends to infinity.

For rational functions with the highest power of the variable of the numerator less than the highest power of the variable of the denominator, the horizontal asymptote is usually given by the equation y = 0.

For rational functions with the highest power of the variable of the numerator equal to the highest power of the variable of the denominator, the horizontal asymptote is usually given by the ratio of the coefficients of the highest power of the variable of the numerator to the coefficient of the highest power of the denominator.

Therefore, the horizontal asymptotes of the function

`f(x)= (3x^2-2x-1)/(x^2+3x-10)`
is

`y= (3)/(1) \\ \\ y=3`