Find all vertical asymptotes of the following function. \[ f(x)=\frac{2 x^{2}-17 x+8}{2 x^{2}-18 x} \]

To find the vertical asymptotes of the function \( f(x) = \frac{2x^{2} - 17x + 8}{2x^{2} - 18x} \), we need to determine where the denominator is equal to zero, as vertical asymptotes occur at these points if the numerator is not also zero at those points. 1. Set the denominator equal to zero: \[ 2x^{2} - 18x = 0 \] Factor out \( 2x \): \[ 2x(x - 9) = 0 \] 2. Set each factor equal to zero: \[ 2x = 0 \quad \Rightarrow \quad x = 0 \\ x - 9 = 0 \quad \Rightarrow \quad x = 9 \] Now we have found potential vertical asymptotes at \( x = 0 \) and \( x = 9 \). Next, we check whether the numerator is equal to zero at these points to ensure they are indeed vertical asymptotes. 3. Evaluate the numerator at \( x = 0 \): \[ 2(0)^{2} - 17(0) + 8 = 8 \quad (\text{not zero}) \] 4. Evaluate the numerator at \( x = 9 \): \[ 2(9)^{2} - 17(9) + 8 = 162 - 153 + 8 = 17 \quad (\text{not zero}) \] Since the numerator does not equal zero at either point, both \( x = 0 \) and \( x = 9 \) are vertical asymptotes. Thus, the vertical asymptotes of the function are: \[ x = 0 \quad \text{and} \quad x = 9. \]