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When graphing rational functions, it's critical to consider the function's overall behavior based on its key features. This involves understanding the relationship between the numerator and the denominator to identify asymptotes and intercepts.

To graph a rational function, identify the x-intercepts by setting the numerator to zero and solving for x, and find the y-intercept by evaluating the function at x = 0. Then, locate the vertical asymptotes by setting the denominator to zero and solving for x, as these lines mark where the function heads towards infinity. As you plot the function within the specified domain, be attentive to the function's end behavior and overall shape, ensuring that regions around asymptotes and intercepts are correctly represented.

Rational functions often involve breaking the graph into distinct regions based on these features. The goal is to create a complete visual representation that displays the function’s behavior, including its increase or decrease, and how it responds as x approaches asymptotes.

Vertical asymptotes are the x-values where a rational function races off to infinity or negative infinity. These occur where the denominator equals zero—since division by zero is undefined, causing a discontinuity in the graph. For our example function, setting each denominator factor to zero, we find vertical asymptotes at x = 4 and x = -1.

However, an asymptote isn't just a boundary—it profoundly affects the function's shape. As x approaches the vertical asymptote values from either side, the function values tend to become extremely large or small, signifying a dramatic 'break' in the graph. They're represented by dashed lines because the function itself never actually reaches those lines. As part of graphing, vertical asymptotes guide you in sketching how the graph behaves near these critical points.

Keep in mind: not every zero of the denominator will be a vertical asymptote if it gets canceled out by the numerator—this leads to a hole in the graph, but that's a topic for another discussion.

Intercepts are the points where the graph of a function crosses the x-axis and y-axis. For x-intercepts, solve for x when the function equals zero (which also means the numerator equals zero). In our function, this gives us an x-intercept at x = 2.

Finding the y-intercept is often simpler—simply evaluate the function with x set to zero. Doing so for our function yields y = -0.5, placing the y-intercept at (0, -0.5). These interceptions mark the points where the graph passes through the axes, anchoring the 'skeleton' of our graph before we draw it in full.

When graphing rational functions, make sure not to confuse an x-intercept for a vertical asymptote; while the x-intercept is a 'crossing' point, the function never touches a vertical asymptote. Mark intercepts clearly on the graph, as they're essential indicators of the function's behavior.

The domain of a function consists of all the input values (x-values) for which the function is defined. For rational functions, any value that makes the denominator zero is not included in the domain, leading to exclusions like x ≠ 4 and x ≠ -1 in our example.

The domain can also be restricted by the context or specific instructions, like how the domain is limited between -3 and 6 in this case.

Range

The range involves all the output values (y-values) the function can take. Determining the range isn't always straightforward but look at how y behaves as x approaches vertical asymptotes, or the domain's maxima and minima. The union of all y-values from these intervals represents the function's range.

Our function’s range is all real numbers because, as x nears the vertical asymptotes, y becomes unbounded. However, any limitations on y due to horizontal asymptotes (not present in this case) or other behavior would restrict the range accordingly, thus completing our understanding of the function’s domain and range.