Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Symmetry in Functions: Multiple Choice Question **Question:** If the graph is that of a function, determine what kinds of symmetry it has. Select all that apply. **Options:** - **A.** The graph is a function and is symmetrical with respect to the y-axis. - **B.** The graph is a function and is symmetrical with respect to the x-axis. - **C.** The graph is a function and is symmetrical with respect to the origin. - **D.** The graph is a function and is not symmetrical. - **E.** The graph is not that of a function. **Explanation:** - **Symmetry with Respect to the y-axis:** A graph is symmetrical with respect to the y-axis if for every point (x, y) on the graph, the point (-x, y) is also on the graph. - **Symmetry with Respect to the x-axis:** A graph is symmetrical with respect to the x-axis if for every point (x, y) on the graph, the point (x, -y) is also on the graph. Note that such a graph usually does not represent a function since it would fail the vertical line test. - **Symmetry with Respect to the Origin:** A graph is symmetrical with respect to the origin if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. - **No Symmetry:** The graph does not exhibit any symmetry. - **Not a Function:** The graph does not represent a function, possibly due to failing the vertical line test. Select the appropriate options based on the given graph of the function.
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Transcribed Image Text:### Symmetry in Functions: Multiple Choice Question **Question:** If the graph is that of a function, determine what kinds of symmetry it has. Select all that apply. **Options:** - **A.** The graph is a function and is symmetrical with respect to the y-axis. - **B.** The graph is a function and is symmetrical with respect to the x-axis. - **C.** The graph is a function and is symmetrical with respect to the origin. - **D.** The graph is a function and is not symmetrical. - **E.** The graph is not that of a function. **Explanation:** - **Symmetry with Respect to the y-axis:** A graph is symmetrical with respect to the y-axis if for every point (x, y) on the graph, the point (-x, y) is also on the graph. - **Symmetry with Respect to the x-axis:** A graph is symmetrical with respect to the x-axis if for every point (x, y) on the graph, the point (x, -y) is also on the graph. Note that such a graph usually does not represent a function since it would fail the vertical line test. - **Symmetry with Respect to the Origin:** A graph is symmetrical with respect to the origin if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. - **No Symmetry:** The graph does not exhibit any symmetry. - **Not a Function:** The graph does not represent a function, possibly due to failing the vertical line test. Select the appropriate options based on the given graph of the function.
**Determining Functions Using the Vertical-Line Test** --- To determine whether a graph represents a function, use the vertical line test. If the graph is a function, use it to find: (a) its domain and range. (b) the intercepts, if any. (c) any symmetry with respect to the x-axis, y-axis, or the origin. | ![Graph](graph-image.png) | |:--:| | This is the graph that needs to be analyzed. | **Is the graph that of a function?** 1. **No** 2. **Yes** [Selected] **If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** A. The domain is ____ . The range is ____ .\ \[Type your answers in interval notation.\] B. The graph is not that of a function. **If the graph is that of a function, what are the intercepts of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** A. The intercept(s) is/are ____ .\ \[Type an ordered pair. Use a comma to separate answers as needed.\] B. The graph is not that of a function. **Description of the Graph** - The graph appears to be plotted on a Cartesian coordinate plane. - The x-axis ranges from -8 to 8, and the y-axis ranges from -4 to 4. - The plotted graph is an ellipse, centered approximately at the origin with major and minor axes along the x and y coordinates, respectively. - The ellipse intersects the x-axis at approximately \((4, 0)\) and \((-4, 0)\) and the y-axis at approximately \((0, 3)\) and \((0, -3)\). To summarize, by using the vertical-line test, we need to check whether any vertical line drawn through the graph touches it at more than one point to determine if it represents a function. If it does represent a function, additional details such as domain, range, and intercepts would be identified and filled out appropriately.
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Transcribed Image Text:**Determining Functions Using the Vertical-Line Test** --- To determine whether a graph represents a function, use the vertical line test. If the graph is a function, use it to find: (a) its domain and range. (b) the intercepts, if any. (c) any symmetry with respect to the x-axis, y-axis, or the origin. | ![Graph](graph-image.png) | |:--:| | This is the graph that needs to be analyzed. | **Is the graph that of a function?** 1. **No** 2. **Yes** [Selected] **If the graph is that of a function, what are the domain and range of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** A. The domain is ____ . The range is ____ .\ \[Type your answers in interval notation.\] B. The graph is not that of a function. **If the graph is that of a function, what are the intercepts of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.** A. The intercept(s) is/are ____ .\ \[Type an ordered pair. Use a comma to separate answers as needed.\] B. The graph is not that of a function. **Description of the Graph** - The graph appears to be plotted on a Cartesian coordinate plane. - The x-axis ranges from -8 to 8, and the y-axis ranges from -4 to 4. - The plotted graph is an ellipse, centered approximately at the origin with major and minor axes along the x and y coordinates, respectively. - The ellipse intersects the x-axis at approximately \((4, 0)\) and \((-4, 0)\) and the y-axis at approximately \((0, 3)\) and \((0, -3)\). To summarize, by using the vertical-line test, we need to check whether any vertical line drawn through the graph touches it at more than one point to determine if it represents a function. If it does represent a function, additional details such as domain, range, and intercepts would be identified and filled out appropriately.
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