Questions: Find all asymptotes of the function f(x)=(9 x^2+6 x-1)/(3 x-2)

Solution Steps

To find the asymptotes of the function \( f(x) = \frac{9x^2 + 6x - 1}{3x - 2} \), we need to consider both vertical and horizontal asymptotes.

1. Vertical Asymptotes: These occur where the denominator is zero. Solve \( 3x - 2 = 0 \) to find the x-values where the function is undefined.
2. Horizontal Asymptotes: These are determined by the degrees of the numerator and the denominator. Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, we look for an oblique asymptote by performing polynomial long division.

To find the vertical asymptotes, we solve for \( x \) where the denominator is zero: \[ 3x - 2 = 0 \] Solving for \( x \): \[ x = \frac{2}{3} \] Thus, the vertical asymptote is at \( x = \frac{2}{3} \).

Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is no horizontal asymptote. Instead, we perform polynomial long division to find the oblique asymptote.

Performing the division of \( 9x^2 + 6x - 1 \) by \( 3x - 2 \), we get: \[ \frac{9x^2 + 6x - 1}{3x - 2} = 3x + 4 + \frac{7}{3x - 2} \] The quotient \( 3x + 4 \) represents the oblique asymptote.

Final Answer