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Understanding the domain of a rational function is crucial because it determines where the function is defined. For the function \( f(x) = \frac{x+2}{x^2-4} \), the domain consists of all real numbers except where the denominator is zero.
To find this, factor the denominator: \[ x^2 - 4 = (x-2)(x+2) \].
This tells us that \( x = 2 \) and \( x = -2 \) are points where the denominator becomes zero, hence these are not in the domain.

• Domain: all real numbers except \( x = 2 \) and \( x = -2 \).
• Discontinuities: these points are discontinuities in the graph; \( x = 2 \) is a vertical asymptote, and \( x = -2 \) is a removable discontinuity.

Recall, removable discontinuities occur when a factor is cancelled from numerator and denominator. Therefore, canceling \( x+2 \) results in a hole at \( x = -2 \) while \( x = 2 \) becomes a vertical asymptote.

Asymptotes provide a guide to the behavior of rational functions. For the simplified function \( g(x) = \frac{1}{x-2} \), analyzing these asymptotes allows us to predict the function's behavior at extreme values.
The vertical asymptote occurs where the denominator equals zero after cancellation of common factors – here, at \( x = 2 \). This is because the function can approach infinity as \( x \) gets closer to this value, but will never actually reach it.
As for the horizontal asymptote, we assess the behavior as \( x \) approaches infinity. Since \( g(x) = \frac{1}{x-2} \) has a constant numerator and the denominator infinitely increases, the value approaches zero.

• Vertical Asymptote: \( x = 2 \)
• Horizontal Asymptote: \( y = 0 \)

Understanding these asymptotes provides a structural frame of reference that will assist in accurately sketching the graph.

Sketching the graph of a rational function involves plotting key features such as asymptotes and discontinuities. For \( g(x) = \frac{1}{x-2} \), begin by marking the vertical asymptote at \( x = 2 \). This is a line the graph approaches but never touches.
Next, the horizontal asymptote at \( y = 0 \) must be sketched, showing the function's value trend as \( x \) increases or decreases towards infinity.
Don’t forget the removable discontinuity at \( x = -2 \) from the original function \( f(x) = \frac{x+2}{x^2-4} \). This is represented in the graph by a hole, or an open circle at \( x = -2 \) on the line.
The curve of \( g(x) \) typically takes the form of a hyperbola, residing in two separate regions split by the vertical asymptote at \( x = 2 \). As it approaches each asymptote, the curve nears but never crosses it.

• Vertical Asymptote: guide for curve placement.
• Horizontal Asymptote: indicative of end behavior.
• Removable Discontinuity: mark with an open circle.

Through these steps, a complete and clear representation of the function’s graph is produced, aiding visualization and understanding.