Final answer:
The question pertains to graphing the function f(x) = x / (x² + 9x) and identifying its characteristics in a high school level mathematics context, which includes finding asymptotes, intercepts, and analyzing the function's behavior to find local maxima, minima, and inflection points.
Step-by-step explanation:
The student's question involves sketching a graph of the function f(x) = x / (x² + 9x) and identifying various characteristics such as domain, asymptotes, intercepts, local maxima and minima, and inflection points. This is a typical high school-level algebra or precalculus exercise. When graphing the function, it's essential to first recognize that the domain excludes values where the denominator equals zero, which in this case is x=0 and x=-9. These values are the vertical asymptotes of the graph.
To find the x-intercept, set the function equal to zero and solve for x, resulting in x=0. There is no y-intercept because the function is undefined at x=0. To find potential local maxima or minima and inflection points, one would calculate the first and second derivatives respectively, and analyze the sign changes.
When sketching the graph, you'll want to calculate these critical points and plot them accordingly. Additionally, observe the behavior of the function as x approaches the vertical asymptotes from both the left and the right to get a more accurate graph. Graphical methods assume that data taken from graphs is accurate to three digits, which helps in estimating values of x and f(x) for various parts of the problem-solving process.