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1 1 2 3 4 x 262 rational functions 1 x2 x 12 domain

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Unformatted text preview: le to fill in any details in those steps which we have abbreviated. Steps for Graphing Rational Functions Suppose r is a rational function. 1. Find the domain of r. 2. Reduce r(x) to lowest terms, if applicable. 3. Find the x- and y -intercepts of the graph of y = r(x), if they exist. 4. Determine the location of any vertical asymptotes or holes in the graph, if they exist. Analyze the behavior of r on either side of the vertical asymptotes, if applicable. 5. Analyze the end behavior of r. Use long division, as needed. 6. Use a sign diagram and plot additional points, as needed, to sketch the graph of y = r(x). 248 Rational Functions 3x . x2 − 4 Solution. We follow the six step procedure outlined above. Example 4.2.1. Sketch a detailed graph of f (x) = 1. As usual, we set the denominator equal to zero to get x2 − 4 = 0. We find x = ±2, so our domain is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞). 2. To reduce f (x) to lowest terms, we factor the numerator and denominator which yields 3x f (x) = (x−2)(x+2) . There are no common factors which means f (x) is already in lowest terms. 3x 3. To find the x-intercepts of the graph of y = f (x), we set y = f (x) = 0. Solving (x−2)(x+2) = 0 results in x = 0. Since x = 0 is in our domain, (0, 0) is the x-intercept. To find the y -intercept, we set x = 0 and find y = f (0) = 0, so that (0, 0) is our y -intercept as well.3 4. The two numbers excluded from the domain of f are x = −2 and x = 2. Since f (x) didn’t reduce...
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