Question

2 which is the correct graph of $y = \frac{2x}{x + 1}$?

Explanation:

Step1: Find vertical asymptote

Set denominator \(x + 1 = 0\), so \(x = -1\) is the vertical asymptote.

Step2: Find horizontal asymptote

Degree of numerator = degree of denominator, so horizontal asymptote is \(y = \frac{2}{1} = 2\).

Step3: Check branches

For \(x > -1\): as \(x\to -1^+\), \(y\to +\infty\); as \(x\to +\infty\), \(y\to 2^+\). For \(x < -1\): as \(x\to -1^-\), \(y\to -\infty\); as \(x\to -\infty\), \(y\to 2^-\). The right graph matches these features.

Answer:

The correct graph is the right one (with two branches separated by \(x = -1\))