To analyze the graph you provided and answer the questions in detail:

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(i) Identify the domain and range:

• Domain: The domain of a graph represents all the possible $x$-values (inputs) that the graph spans. To determine this:

  • Observe the $x$-axis and identify the smallest and largest $x$-values covered by the graph.
  • If the graph continues indefinitely in the horizontal direction, its domain could be all real numbers ($\mathbb{R}$).
  • If it stops at specific points, the domain is restricted to those $x$-values.

• Range: The range of a graph represents all the possible $y$-values (outputs) that the graph covers. To find this:

  • Examine the $y$-axis and determine the lowest and highest points reached by the graph.

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(ii) Does this graph represent a function and a one-one function? Why/Why not?

• Function:

  • A graph represents a function if, for every $x$-value in the domain, there is exactly one $y$-value. This can be verified using the Vertical Line Test:
    • Draw imaginary vertical lines at various $x$-values. If any vertical line intersects the graph more than once, the graph does not represent a function.

• One-One Function:

  • A graph represents a one-one function if every $x$-value has a unique $y$-value, and no two $x$-values share the same $y$-value. This can be tested using the Horizontal Line Test:
    • Draw imaginary horizontal lines. If any horizontal line intersects the graph more than once, the graph is not one-one.

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Detailed Explanation/Justification:

• If the graph represents a function, it must pass the vertical line test. For example:
  • If a vertical line intersects the graph at two or more points, this implies multiple outputs for the same input, violating the definition of a function.
• For a one-one function, it must also pass the horizontal line test. For example:
  • If a horizontal line intersects the graph at multiple points, this indicates that multiple inputs share the same output, which disqualifies it as one-one.

Please share the graph for a more specific interpretation. If the graph is unavailable, consider describing its shape or features for further assistance.

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Do you have any questions or want further clarification?

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5 Related Questions:

1. How can we apply the domain and range to solve real-world problems?
2. What is the significance of the vertical and horizontal line tests in analyzing functions?
3. Can a graph fail to represent a function but still have a defined domain and range?
4. How does the concept of one-one functions relate to invertibility of functions?
5. How does restricting the domain affect the range and the property of being one-one?

Tip: Always start analyzing a graph by identifying its critical points, such as intercepts, maxima, minima, and asymptotes. This provides key insights into its behavior.