Graph all vertical and horizontal asymptotes of the rational function. \[ f(x)=\frac{-6 x-7}{4 x-6} \]

Function by following steps: - step0: Find the horizontal asymptotes: \(f\left(x\right)=-6x-\frac{7}{4x-6}\) - step1: Evaluate the limits \(\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)\) and \(\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right):\) \(\begin{align}&\lim _{x\rightarrow +\infty}\left(-6x-\frac{7}{4x-6}\right)\\&\lim _{x\rightarrow -\infty}\left(-6x-\frac{7}{4x-6}\right)\end{align}\) - step2: Calculate: \(\begin{align}&-\infty\\&+\infty\end{align}\) - step3: The function has no horizontal asymptotes: \(\textrm{No horizontal asymptotes}\) Find the vertical asymptotes of \( f(x)=-6*x-7/(4*x-6) \). Function by following steps: - step0: Find the vertical asymptotes: \(f\left(x\right)=-6x-\frac{7}{4x-6}\) - step1: Evaluate the limit: \(\lim _{x\rightarrow \frac{3}{2}}\left(-6x-\frac{7}{4x-6}\right)\) - step2: Evaluate the left-hand and the right-hand limits: \(\begin{align}&\lim _{x\rightarrow \frac{3}{2}^{-}}\left(-6x-\frac{7}{4x-6}\right)\\&\lim _{x\rightarrow \frac{3}{2}^{+}}\left(-6x-\frac{7}{4x-6}\right)\end{align}\) - step3: Evaluate the left-hand limit: \(\begin{align}&+\infty\\&-\infty\end{align}\) - step4: Limit does not exist: \(\textrm{The limit does not exist}\) - step5: \(x=\frac{3}{2}\) is a vertical asymptote\(:\) \(\begin{align}&x=\frac{3}{2}\textrm{ }\textrm{is a vertical asymptote}\end{align}\) - step6: List all vertical asymptotes of the function: \(\begin{align}&x=\frac{3}{2}\end{align}\) To analyze the rational function \[ f(x) = \frac{-6x - 7}{4x - 6} \] we need to find its vertical and horizontal asymptotes. ### Vertical Asymptotes Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points). Setting the denominator equal to zero: \[ 4x - 6 = 0 \] Solving for \(x\): \[ 4x = 6 \implies x = \frac{3}{2} \] Thus, there is a vertical asymptote at \[ x = \frac{3}{2}. \] ### Horizontal Asymptotes Horizontal asymptotes are determined by the degrees of the polynomial in the numerator and the denominator. In this case, both the numerator and denominator are linear polynomials (degree 1). For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator \(-6x\) is \(-6\) and the leading coefficient of the denominator \(4x\) is \(4\). Therefore, the horizontal asymptote is: \[ y = \frac{-6}{4} = -\frac{3}{2}. \] ### Summary of Asymptotes - **Vertical Asymptote:** \(x = \frac{3}{2}\) - **Horizontal Asymptote:** \(y = -\frac{3}{2}\) These asymptotes can be graphed on the coordinate plane to visualize the behavior of the function.