Question

choose the function represented by this graph: select one: a. $f(x) = \frac{x^3 - 1}{x + 1}$ b. $f(x) = \frac{x^2 - 10x + 1}{x + 2}$ c. $f(x) = \frac{x^3}{x + 2}$ d. $f(x) = \frac{x + 1}{x^3 + 2}$

Explanation:

Step1: Check domain of the graph

The graph has a vertical asymptote at x=-2 (undefined at x=-2), so denominator has factor (x+2). Options a (denominator x+1), d (denominator x²+2, no real asymptote) are eliminated. Remaining: b, c.

Step2: Simplify option a (for check)

Option a: $\frac{x^3 -1}{x+1} = \frac{(x-1)(x²+x+1)}{x+1}$, asymptote at x=-1, not matching graph.

Step3: Analyze end behavior of b and c

Option b: Degree of numerator (2) < denominator (1)? No, numerator degree 2, denominator 1: end behavior is linear (like ax + b). Option c: Numerator degree 3, denominator 1: end behavior is quadratic (curved), but graph’s right side is linear. So option b is linear-like, matching the graph.

Answer:

b. $f(x) = \frac{x^2 -10x +1}{x+2}$