Short Answer 
The function (f(x) = dfrac{2x}{x^2 – 1}) is a rational function with vertical asymptotes at (x = 1) and (x = -1) and passes through the origin. To sketch its graph, plot the origin, indicate the asymptotes, and capture the hyperbolic behavior in the first and third quadrants.
Step 1: Understand the Function’s Form
The given function is a rational function represented as (f(x) = dfrac{2x}{x^2 – 1}). This type of function is created by dividing two polynomials. In this expression, the numerator is (2x) and the denominator is (x^2 – 1). To analyze its behavior, we should recognize the characteristics of both the polynomial terms involved and identify any vertical asymptotes where the denominator equals zero.
Step 2: Identify Key Properties
By examining the function further, you can determine certain properties: It clearly passes through the origin (0, 0) since substituting (x = 0) yields (f(0) = 0). The behavior in the quadrants can be inferred from the function’s structure; specifically, the hyperbola appears in the first and third quadrants as determined by the signs of the function in those areas. The vertical asymptotes occur at points where the denominator is zero, i.e., at (x = 1) and (x = -1).
Step 3: Sketch the Function’s Graph
To sketch the graph of this function, follow these guidelines:
- Plot the point (0, 0) as it lies on the curve.
- Indicate the vertical asymptotes at (x = 1) and (x = -1), where the function approaches infinity.
- Utilize the behavior of the hyperbola to extend curves towards the asymptotes while ensuring they are in the first and third quadrants.
- Include a cubic-like behavior by curving downwards after passing through the origin.