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Domain: (-∞, -5) U (-5,3) U (3, 00) | *x-intercepts: (-7,0) and (2,0) (and no others) *y-intercept: (0, 3) *Horizontal Asymptote: y = 4 dictating tail-end behavior as a too *Vertical Asymptotes: r = -5 and x = 3 (and no others) %3D f is decreasing on (-0o, -5), (-5, 3), and (3, 0) and increasing nowhere •f is concave downward on (-o,-5) and (-1,3), and concave upward on (-5, -1) and (3, 00). *Inflection Point: (-1,5) |

Domain: (-∞, -5) U (-5,3) U (3, 00) | *x-intercepts: (-7,0) and (2,0) (and no others) *y-intercept: (0, 3) *Horizontal Asymptote: y = 4 dictating tail-end behavior as a too *Vertical Asymptotes: r = -5 and x = 3 (and no others) %3D f is decreasing on (-0o, -5), (-5, 3), and (3, 0) and increasing nowhere •f is concave downward on (-o,-5) and (-1,3), and concave upward on (-5, -1) and (3, 00). *Inflection Point: (-1,5) |

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Sketch the graph of a single function

### Mathematical Analysis of Function Characteristics This analysis provides a detailed overview of a mathematical function based on its domain, intercepts, asymptotes, intervals of increase/decrease, concavity, and inflection points. --- #### Key Characteristics: - **Domain:** The function is defined for the intervals \((-\infty, -5) \cup (-5, 3) \cup (3, \infty)\). - **\(x\)-intercepts:** The points where the function crosses the x-axis are \((-7, 0)\) and \((2, 0)\). There are no other \(x\)-intercepts. - **\(y\)-intercept:** The function crosses the y-axis at the point \((0, 3)\). - **Horizontal Asymptote:** As \(x\) approaches \(\pm \infty\), the function approaches the line \(y = 4\). - **Vertical Asymptotes:** The function has vertical asymptotes at \(x = -5\) and \(x = 3\). There are no other vertical asymptotes. - **Intervals of Decrease:** - The function is decreasing on the intervals \((-\infty, -5)\), \((-5, 3)\), and \((3, \infty)\). - The function is not increasing on any interval. - **Concavity:** - The function is concave downward on the intervals \((-\infty, -5)\) and \((-1, 3)\). - It is concave upward on the interval \((-5, -1)\). - **Inflection Point:** The function has an inflection point at \((-1, 5)\). --- #### Graph Description: The accompanying graph shows a standard Cartesian coordinate system with labeled axes \(x\) and \(y\). The major points, intervals, and asymptotic behavior as described above are expected to be represented on this graph to illustrate the function's behavior visually. This analysis would be suitable for educational purposes to help students understand function behaviors including asymptotes, intercepts, and changes in concavity.
Transcribed Image Text:### Mathematical Analysis of Function Characteristics This analysis provides a detailed overview of a mathematical function based on its domain, intercepts, asymptotes, intervals of increase/decrease, concavity, and inflection points. --- #### Key Characteristics: - **Domain:** The function is defined for the intervals \((-\infty, -5) \cup (-5, 3) \cup (3, \infty)\). - **\(x\)-intercepts:** The points where the function crosses the x-axis are \((-7, 0)\) and \((2, 0)\). There are no other \(x\)-intercepts. - **\(y\)-intercept:** The function crosses the y-axis at the point \((0, 3)\). - **Horizontal Asymptote:** As \(x\) approaches \(\pm \infty\), the function approaches the line \(y = 4\). - **Vertical Asymptotes:** The function has vertical asymptotes at \(x = -5\) and \(x = 3\). There are no other vertical asymptotes. - **Intervals of Decrease:** - The function is decreasing on the intervals \((-\infty, -5)\), \((-5, 3)\), and \((3, \infty)\). - The function is not increasing on any interval. - **Concavity:** - The function is concave downward on the intervals \((-\infty, -5)\) and \((-1, 3)\). - It is concave upward on the interval \((-5, -1)\). - **Inflection Point:** The function has an inflection point at \((-1, 5)\). --- #### Graph Description: The accompanying graph shows a standard Cartesian coordinate system with labeled axes \(x\) and \(y\). The major points, intervals, and asymptotic behavior as described above are expected to be represented on this graph to illustrate the function's behavior visually. This analysis would be suitable for educational purposes to help students understand function behaviors including asymptotes, intercepts, and changes in concavity.
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