We will analyze the rational function f(x) = 2x over (x minus 2). This is a ratio of two polynomials.
To find the domain, we must exclude values that make the denominator zero. Set denominator to zero: x minus 2 equals 0, so x equals 2. Domain is all real numbers except x equals 2.
As x approaches 2, the function approaches infinity. The limit as x approaches 2 from the left is negative infinity, and from the right is positive infinity. This creates a vertical asymptote at x equals 2.
To find the horizontal asymptote, we examine end behavior. Divide numerator and denominator by x, giving f(x) equals 2 over (1 minus 2 over x). As x approaches positive or negative infinity, 2 over x approaches 0, so f(x) approaches 2. Horizontal asymptote at y equals 2.
Find intercepts and key points. y-intercept: f(0) equals 0 over negative 2 equals 0, point (0, 0). x-intercept: 2x equals 0, so x equals 0, point (0, 0). Key points: f(1) equals negative 2, point (1, negative 2); f(3) equals 6, point (3, 6); f(4) equals 4, point (4, 4).
Combining all features: vertical asymptote at x equals 2, horizontal asymptote at y equals 2, passes through origin (0, 0), and key points plotted. The two branches approach the asymptotes but never touch.
Function behavior summary: Decreasing on negative infinity to 2 and 2 to infinity. Approaches vertical asymptote: as x approaches 2 from the left, f(x) approaches negative infinity; as x approaches 2 from the right, f(x) approaches positive infinity. Approaches horizontal asymptote: as x approaches positive or negative infinity, f(x) approaches 2. Range is negative infinity to 2 union 2 to infinity.