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: