The problem asks to determine the values of $A$, $B$, $C$, and $D$ for the function $g(x) = 2f(x) + 5$, based on the provided table for $f(x)$.

The values of $f(x)$ at different points $x = -2, 0, 1, 5$ are given, and we are asked to compute the corresponding values of $g(x)$ using the equation $g(x) = 2f(x) + 5$.

Step-by-step Calculation:

1. For $x = -2$:

   • $f(-2) = -4$
   • $g(-2) = 2f(-2) + 5 = 2(-4) + 5 = -8 + 5 = -3$
   • Therefore, $A = -3$.

2. For $x = 0$:

   • $f(0) = -1$
   • $g(0) = 2f(0) + 5 = 2(-1) + 5 = -2 + 5 = 3$
   • Therefore, $B = 3$.

3. For $x = 1$:

   • $f(1) = 3$
   • $g(1) = 2f(1) + 5 = 2(3) + 5 = 6 + 5 = 11$
   • Therefore, $C = 11$.

4. For $x = 5$:

   • $f(5) = 7$
   • $g(5) = 2f(5) + 5 = 2(7) + 5 = 14 + 5 = 19$
   • Therefore, $D = 19$.

Final Answers:

• $A = -3$
• $B = 3$
• $C = 11$
• $D = 19$

Would you like a more detailed explanation on any step? Here are some related questions for further practice:

1. If $g(x) = 3f(x) + 4$, what would $A$, $B$, $C$, and $D$ be?
2. What happens to $g(x)$ if $f(x)$ is multiplied by a different constant?
3. Can you describe how the graph of $g(x)$ relates to the graph of $f(x)$?
4. How would you solve if the function $g(x)$ was given as $g(x) = f(x) - 2$?
5. If $g(x)$ was a quadratic function, what would the new values of $A$, $B$, $C$, and $D$ be?

Tip: When modifying functions, always check if a transformation affects the slope, the intercept, or both.