https://file.aiquickdraw.com/mathbot/file/1726958596847jlsn3yz7.jpg

To determine the piecewise-defined function based on the given graph, let's break it down by analyzing the behavior of the function over specific intervals.

### Key Points on the Graph:
1. **At \( x = -1 \)**: The point is \( (-1, -1) \), and there is a curve passing through it.
2. **At \( x = 0 \)**: The point is \( (0, 0) \).
3. **At \( x = 1 \)**: The point is \( (1, \frac{1}{3}) \).
4. **At \( x = 9 \)**: The point is \( (9, 3) \).
5. **At \( x = 10 \)**: The point is \( (10, 7) \).

The graph is composed of different segments that change at specific values of \( x \), indicating a piecewise function.

### Step-by-Step Breakdown:

#### For \( x \leq -1 \):
- The leftmost part of the graph seems to be a horizontal line or a constant value near \( y = -1 \). This suggests that:
  \[
  f(x) = -1 \quad \text{for} \quad x \leq -1.
  \]

#### For \( -1 < x \leq 1 \):
- The graph appears to be a smooth curve passing through the points \( (-1, -1) \), \( (0, 0) \), and \( (1, \frac{1}{3}) \). This suggests a quadratic or cubic relationship. Based on these values, the equation of this part of the curve can be approximated as:
  \[
  f(x) = x^2 - 1 \quad \text{for} \quad -1 < x \leq 1.
  \]

#### For \( 1 < x \leq 9 \):
- Between \( x = 1 \) and \( x = 9 \), the graph is linear, connecting the points \( (1, \frac{1}{3}) \) and \( (9, 3) \). The slope of the line is:
  \[
  \text{slope} = \frac{3 - \frac{1}{3}}{9 - 1} = \frac{\frac{8}{3}}{8} = \frac{1}{3}.
  \]
  Thus, the equation of this line is:
  \[
  f(x) = \frac{1}{3}(x - 1) + \frac{1}{3} \quad \text{for} \quad 1 < x \leq 9.
  \]

#### For \( x = 10 \):
- At \( x = 10 \), the graph shows a point \( (10, 7) \). Since this is an isolated point, the value of the function here is:
  \[
  f(x) = 7 \quad \text{for} \quad x = 10.
  \]

### Answer:
The piecewise function describing this graph is:

\[
f(x) =
\begin{cases}
-1 & \text{if} \quad x \leq -1 \\
x^2 - 1 & \text{if} \quad -1 < x \leq 1 \\
\frac{1}{3}(x - 1) + \frac{1}{3} & \text{if} \quad 1 < x \leq 9 \\
7 & \text{if} \quad x = 10
\end{cases}
\]

Would you like me to explain any specific step in further detail or help with related questions?

Here are 5 related questions to deepen your understanding:
1. How do you determine the equation of a line given two points?
2. What is a piecewise function, and how is it different from a regular function?
3. Can a function have an isolated point like \( f(10) = 7 \)? How is this represented graphically?
4. How do you determine if a graph represents a linear, quadratic, or cubic function?
5. What is the process of constructing the equation of a curve passing through multiple points?

**Tip:** When analyzing graphs for piecewise functions, always look for key transition points (like turning points or changes in slope), and use these points to break the function into manageable pieces.

Find the rule that describes the piecewise-defined function based on the graph provided. The graph shows different segments with specific points, including (-1, -1), (0, 0), (1, 1/3), (9, 3), and (10, 7).

Learn how to determine the piecewise-defined function from a graph showing linear and quadratic segments with specific points such as (-1, -1) and (10, 7). Suitable for Grades 9-12, this lesson covers key concepts like linear equations, quadratic functions, and analyzing graphs.

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