Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x³ - 6x² + 9x
5. Graphical Applications of Derivatives
Curve Sketching
9:56 minutesProblem 4.4.44
Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 1/(e⁻ˣ - 1)
Verified step by step guidance1
Identify the domain of the function f(x) = \(\frac{1}{e^{-x} - 1}\). The expression e^{-x} - 1 cannot be zero, so solve e^{-x} - 1 = 0 to find the values of x that are not in the domain.
Determine the vertical asymptotes by finding the values of x where the denominator is zero. Since e^{-x} = 1 when x = 0, there is a vertical asymptote at x = 0.
Analyze the behavior of the function as x approaches the vertical asymptote from both sides. As x approaches 0 from the left, e^{-x} approaches infinity, making the denominator approach infinity, and thus f(x) approaches 0. As x approaches 0 from the right, e^{-x} approaches 1, making the denominator approach 0, and thus f(x) approaches infinity.
Examine the end behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, e^{-x} approaches 0, making the denominator approach -1, so f(x) approaches -1. As x approaches negative infinity, e^{-x} approaches infinity, making the denominator approach infinity, so f(x) approaches 0.
Identify any horizontal asymptotes based on the end behavior. From the analysis, as x approaches positive infinity, f(x) approaches -1, indicating a horizontal asymptote at y = -1. As x approaches negative infinity, f(x) approaches 0, indicating a horizontal asymptote at y = 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) = 1/(e⁻ˣ - 1), the domain excludes values where the denominator is zero, i.e., e⁻ˣ - 1 = 0, which occurs at x = 0. Thus, the domain is all real numbers except x = 0.
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Asymptotes
Asymptotes are lines that a graph approaches but never touches. For the function f(x) = 1/(e⁻ˣ - 1), there is a vertical asymptote at x = 0, where the function is undefined. Additionally, as x approaches positive or negative infinity, the function approaches horizontal asymptotes, which can be determined by analyzing the behavior of the function as x becomes very large or very small.
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Behavior at Infinity
Understanding the behavior of a function as x approaches infinity or negative infinity helps in sketching its graph. For f(x) = 1/(e⁻ˣ - 1), as x approaches positive infinity, e⁻ˣ approaches zero, making the function approach -1. As x approaches negative infinity, e⁻ˣ becomes very large, making the function approach zero. This behavior helps in identifying horizontal asymptotes and the overall shape of the graph.
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