Question

which graph represents the function $f(x) = \frac{2x}{x^2 - 1}$?

Explanation:

Step1: Analyze the function's properties

The function is \( f(x)=\frac{2x}{x^{2}-1}=\frac{2x}{(x - 1)(x + 1)} \). The domain is all real numbers except \( x = 1 \) and \( x=- 1 \) (vertical asymptotes at \( x = 1 \) and \( x=-1 \)). Also, check if the function is odd: \( f(-x)=\frac{-2x}{x^{2}-1}=-f(x) \), so it's an odd function, symmetric about the origin.

Step2: Analyze the graphs

• The third graph is even (symmetric about y - axis), so eliminate it.
• The first graph: Let's check the behavior around \( x = 0 \), \( f(0)=0 \). For \( x>1 \), say \( x = 2 \), \( f(2)=\frac{4}{4 - 1}=\frac{4}{3}>0 \), the right part should be above x - axis. For \( - 1
• The second graph: Check symmetry (odd, symmetric about origin). For \( x>1 \), \( f(x)>0 \) (as \( x = 2 \) gives positive), for \( - 1
• The fourth graph: For \( x>1 \), \( f(x)>0 \), but the right part seems to have a different trend (maybe decreasing too fast or wrong sign? No, the second graph is better.

Answer:

The Middle Graph (the second graph among the three top - row graphs or the one in the middle of the initial three - graph layout)