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Find the domain, range and sketch the graph of the function f(x)=3x−6/x-3 through the asymptotes and the x,y-intercept points.

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Final answer:

The domain of the function is (-∞, 3) U (3, ∞). The range of the function is (-∞, -3) U (3, ∞). The graph of the function has a vertical asymptote at x = 3 and passes through the x-intercept (2, 0) and the y-intercept (0, -2).

Step-by-step explanation:

To find the domain of the function f(x) = 3x - 6 / x - 3, we need to determine the values of x that make the denominator, x - 3, equal to zero. So, x cannot be equal to 3. Therefore, the domain of the function is all real numbers except 3, which can be written as (-∞, 3) U (3, ∞).

To find the range of the function, we need to analyze the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, the function approaches 3, and as x approaches negative infinity, the function approaches -3. Therefore, the range of the function is (-∞, -3) U (3, ∞).

To sketch the graph of the function, we can use the information about the domain, range, and asymptotes. The vertical asymptote is x = 3, and there is no horizontal asymptote. The x-intercept can be found by setting y = 0 and solving for x, which gives x = 2. The y-intercept can be found by setting x = 0, which gives y = -2. Using this information, we can plot points on the graph and draw a curve that approaches the asymptote x = 3 as x approaches 3.

Learn more about Graphing Rational Functions

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