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