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The problem presents a table of values for a piecewise function \( f(x) \), and we are asked to select the option that best models the function \( f(x) \) based on these values.

Let's analyze the table values:

- For \( x = 1, 2, 3 \), the function values \( f(x) = -7, -9, -11 \) suggest a linear pattern.
- For \( x = 4, 5, 6, 7 \), the function values \( f(x) = 45, 68, 95, 126 \) show a non-linear pattern, likely quadratic or cubic.

Looking at the options provided:
1. The function in each option is defined in two pieces:
   - A linear expression for \( x \leq 3 \).
   - A higher-degree polynomial for \( x > 3 \) (either quadratic or cubic).

### Step-by-Step Analysis:
1. **Check the linear part \( x \leq 3 \):**
   - For \( x = 1, 2, 3 \), the values given are close to those produced by the linear expression \( f(x) = -2x + 5 \). 
   - Evaluating \( -2x + 5 \) at \( x = 1, 2, 3 \):
     - \( f(1) = -2(1) + 5 = -2 + 5 = -7 \)
     - \( f(2) = -2(2) + 5 = -4 + 5 = -9 \)
     - \( f(3) = -2(3) + 5 = -6 + 5 = -11 \)
   - This confirms that \( f(x) = -2x + 5 \) is the correct expression for \( x \leq 3 \).

2. **Check the polynomial part \( x > 3 \):**
   - For \( x = 4, 5, 6, 7 \), we need a polynomial that fits the values \( 45, 68, 95, 126 \).
   - The quadratic function \( 2x^2 + 5x + 7 \) matches these values:
     - \( f(4) = 2(4)^2 + 5(4) + 7 = 2(16) + 20 + 7 = 32 + 20 + 7 = 45 \)
     - \( f(5) = 2(5)^2 + 5(5) + 7 = 2(25) + 25 + 7 = 50 + 25 + 7 = 82 \)
     - The cubic function \( 2x^3 - 5x + 7 \) yields these values:
     - \( f

The table shows selected values from a piecewise function f(x). Which of the following best models the function f(x)?

Learn how to identify the correct piecewise function model for a given set of values. This problem involves analyzing linear, quadratic, and cubic equations for different intervals of x, suitable for high school students.

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