Question

Which graph represents the function f (x) = StartFraction 2 x Over x squared minus 1 EndFraction?
On a coordinate plane, a hyperbola and parabola are shown. The hyperbola has a curve in quadrants 1 and 2. The parabola opens down and has a vertex at (0, 0).
On a coordinate plane, a hyperbola and a cubic functions are shown. They hyperbola has curves in quadrants 1 and 3, and the cubic function goes through (0, 0).
On a coordinate plane, a hyperbola is shown. One curve is in quadrant 1 and the other curves goes through quadrants 2 and 4.
On a coordinate plane, a hyperbola is shown. Both curves open downward in quadrants 3 and 4.

Answer 1

Choose the third graph.

Reason: f(x)=2x/(x^2−1) has vertical asymptotes at x=±1 and horizontal asymptote y=0. f(0)=0, and f is odd, so there is a continuous branch on (−1,1) passing through (0,0) (it runs through quadrants II and IV), and outer branches for x>1 and x<−1 lie in quadrants I and III. The third description matches this shape.