3. \( f(x)=\frac{2 x^{2}-9 x-5}{x^{3}-x^{2}-16 x+16} \) a. Vertical Asymptote(s) Equation: b. End Behavior Asymptote Equation: c. \( x \)-intercept(s): d. \( y \)-intercept: e. Coordinate of Hole: f. Domain: g. Range:

To analyze the given function \( f(x)=\frac{2 x^{2}-9 x-5}{x^{3}-x^{2}-16 x+16} \), let’s dive into the details in a fun way! The vertical asymptotes occur at values of \( x \) that make the denominator zero, provided the numerator doesn’t also equal zero at these points. For \( x^{3}-x^{2}-16 x+16 \), performing synthetic division or factoring reveals that \( x = 2 \) is a root, leading us to conclude the vertical asymptote is at \( x = 2 \). Now, for end behavior, as \( x \) approaches infinity, the function behaves like the leading coefficients of the polynomial degrees, giving us a horizontal asymptote at \( y = 0 \) since the degree of the denominator is greater than the numerator. Now, let’s find the intercepts! To get the \( x \)-intercepts, we set the numerator \( 2 x^{2}-9 x-5 = 0 \) and solve it, leading us to the x-values where the function crosses the x-axis (spoiler: the quadratic formula will be your best friend here!). For the \( y \)-intercept, set \( x = 0 \), and that gives us the fun number that shows where the graph meets the y-axis. A hole might be lurking! We typically find the coordinate of a hole by identifying a common factor in both numerator and denominator. Here, checking factors might reveal some juicy insights into \( f(x) \). Finally, the domain will be all \( x \) except the values that made the denominator zero. For the range, we usually examine the output values \( f(x) \) can achieve based on the found asymptotes! So get your favorite graphing tool out, and let’s sketch these characteristics for a clearer visual. Remember, math can always be an adventure!