Questions: Sketch the graph of the rational function. f(x) = 2x / (x^2 - 3x - 4)

Transcript text: Sketch the graph of the rational function. \[ f(x)=\frac{2 x}{x^{2}-3 x-4} \]

Solution

Solution Steps

Step 1: Identify the function

The given function is: \[ f(x) = \frac{2x}{x^2 - 3x - 4} \]

Step 2: Find the vertical asymptotes

The vertical asymptotes occur where the denominator is zero: \[ x^2 - 3x - 4 = 0 \] Solving for $ x $: \[ (x - 4)(x + 1) = 0 \] Thus, the vertical asymptotes are at: \[ x = 4 \] \[ x = -1 \]

Step 3: Find the horizontal asymptote

For large values of $ x $, the degree of the numerator (1) is less than the degree of the denominator (2), so the horizontal asymptote is: \[ y = 0 \]

Final Answer

The function $ f(x) = \frac{2x}{x^2 - 3x - 4} $ has vertical asymptotes at $ x = 4 $ and $ x = -1 $, and a horizontal asymptote at $ y = 0 $.

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["y = (2x)/(x**2 - 3x - 4)"], "latex_expressions": ["$f(x) = \\frac{2x}{x^2 - 3x - 4}$"]}

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