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Select all of the following graphs which are one-to-one functi

Select all of the following graphs which are one-to-one functi

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This image contains four separate graphs, each demonstrating different types of functions on a Cartesian coordinate system. 1. **Graph 1: Linear Function** - Description: The graph displays a straight line with a negative slope, indicating a linear function. It crosses the y-axis above the origin and extends downward as it moves to the right. - Characteristics: The line steadily decreases, indicating a constant rate of change. 2. **Graph 2: Quadratic Function** - Description: This graph shows a parabola opening upwards. The curve reaches its minimum point at the vertex and then rises symmetrically on both sides. - Characteristics: The function has one minimum point, representing a positive quadratic coefficient. 3. **Graph 3: Trigonometric Function** - Description: The graph illustrates a sinusoidal wave, typically representative of a sine or cosine function. The wave oscillates between positive and negative values, with peaks and troughs symmetrically distributed. - Characteristics: The amplitude and period are consistent, showing the periodic nature of trigonometric functions. 4. **Graph 4: Logarithmic or Exponential Function** - Description: A curve starting from just below the y-axis and extending upwards with a rapid increase, indicating either a logarithmic or exponential nature. - Characteristics: The rate of increase becomes less steep over time, often seen in growth or decay models. At the bottom of the image, there are interactive elements for user engagement: - **Question Help:** An option to access a video for additional guidance. - **Submit Question:** A button likely used for students to submit responses or inquiries regarding the graphs. This educational visualization aids in understanding different mathematical functions and their graphical representations.
Transcribed Image Text:This image contains four separate graphs, each demonstrating different types of functions on a Cartesian coordinate system. 1. **Graph 1: Linear Function** - Description: The graph displays a straight line with a negative slope, indicating a linear function. It crosses the y-axis above the origin and extends downward as it moves to the right. - Characteristics: The line steadily decreases, indicating a constant rate of change. 2. **Graph 2: Quadratic Function** - Description: This graph shows a parabola opening upwards. The curve reaches its minimum point at the vertex and then rises symmetrically on both sides. - Characteristics: The function has one minimum point, representing a positive quadratic coefficient. 3. **Graph 3: Trigonometric Function** - Description: The graph illustrates a sinusoidal wave, typically representative of a sine or cosine function. The wave oscillates between positive and negative values, with peaks and troughs symmetrically distributed. - Characteristics: The amplitude and period are consistent, showing the periodic nature of trigonometric functions. 4. **Graph 4: Logarithmic or Exponential Function** - Description: A curve starting from just below the y-axis and extending upwards with a rapid increase, indicating either a logarithmic or exponential nature. - Characteristics: The rate of increase becomes less steep over time, often seen in growth or decay models. At the bottom of the image, there are interactive elements for user engagement: - **Question Help:** An option to access a video for additional guidance. - **Submit Question:** A button likely used for students to submit responses or inquiries regarding the graphs. This educational visualization aids in understanding different mathematical functions and their graphical representations.
## Identifying One-to-One Functions ### Instructions Select all of the following graphs which are **one-to-one functions**. ### Graph Descriptions 1. **Graph 1:** - This graph features a curve that starts at the bottom left and rises towards the right before looping back down. The curve fails the horizontal line test at several points, indicating it is not a one-to-one function. 2. **Graph 2:** - The graph shows an ellipse centered around the point (0, -2) on the coordinate plane. An ellipse encapsulates multiple y-values for a single x-value, which does not pass the horizontal line test, making it not a one-to-one function. 3. **Graph 3:** - A straight diagonal line runs from the top left to the bottom right. The linearity of this graph ensures that it passes the horizontal line test, qualifying it as a one-to-one function. 4. **Graph 4:** - This graph depicts a parabola opening upwards, starting from the far left and rising towards the vertex near the top center. The parabola fails the horizontal line test at several y-values, making it not a one-to-one function. ### Conclusion To determine if a graph represents a one-to-one function, apply the horizontal line test: a function is one-to-one if no horizontal line intersects the graph at more than one point. Based on the graph descriptions, only the straight line in Graph 3 is a one-to-one function.
Transcribed Image Text:## Identifying One-to-One Functions ### Instructions Select all of the following graphs which are **one-to-one functions**. ### Graph Descriptions 1. **Graph 1:** - This graph features a curve that starts at the bottom left and rises towards the right before looping back down. The curve fails the horizontal line test at several points, indicating it is not a one-to-one function. 2. **Graph 2:** - The graph shows an ellipse centered around the point (0, -2) on the coordinate plane. An ellipse encapsulates multiple y-values for a single x-value, which does not pass the horizontal line test, making it not a one-to-one function. 3. **Graph 3:** - A straight diagonal line runs from the top left to the bottom right. The linearity of this graph ensures that it passes the horizontal line test, qualifying it as a one-to-one function. 4. **Graph 4:** - This graph depicts a parabola opening upwards, starting from the far left and rising towards the vertex near the top center. The parabola fails the horizontal line test at several y-values, making it not a one-to-one function. ### Conclusion To determine if a graph represents a one-to-one function, apply the horizontal line test: a function is one-to-one if no horizontal line intersects the graph at more than one point. Based on the graph descriptions, only the straight line in Graph 3 is a one-to-one function.
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