https://file.aiquickdraw.com/mathbot/file/1727472411674mirt5b23.jpg

Let's analyze the image you've provided. The task is to determine which of the given tables correctly reflects the function \( f(x) = 2^x + x^2 \).

To verify which table is correct, we will compute \( f(x) = 2^x + x^2 \) for several values of \( x \) and compare them with the values shown in the tables. Here's how we calculate \( f(x) \):

1. \( f(x) = 2^x + x^2 \) for each value of \( x \) in the table.
2. Compare the computed values to the values provided in each option (a, b, c, d).

Let's compute the values of \( f(x) \) for the integers \( x = -3, -2, -1, 0, 1, 2, 3, 4 \).

I'll calculate them now.The error occurred because raising 2 to a negative power is not directly supported in this context when using integers. To fix this, I'll adjust the computation to handle negative exponents correctly.

I'll recalculate \( f(x) = 2^x + x^2 \) while ensuring that fractional values are handled for negative powers of \( x \).The computed values for \( f(x) = 2^x + x^2 \) are:

\[
f(x) = [9.125, 4.25, 1.5, 1.0, 3.0, 8.0, 17.0, 32.0]
\]

Now, let's compare these results with the values in the tables:

- **Option A**:  
  \( f(x) = [9.1, 4.3, 1.5, 1.0, 3.0, 8.0, 17.0, 32.0] \)
  
  This is very close to our computed values, with small rounding differences, particularly for \( x = -3 \) and \( x = -2 \).

- **Option B**:  
  \( f(x) = [-8.9, -3.8, -0.5, 1.0, 3.0, 8.0, 17.0, 32.0] \)
  
  The values for negative \( x \) do not match our computed results.

- **Option C**:  
  \( f(x) = [-15.0, -8.0, -3.0, 1.0, 3.0, 8.0, 17.0, 32.0] \)
  
  These values also do not match for negative \( x \).

- **Option D**:  
  \( f(x) = [3.0, 0.0, -1.0, 1.0, 3.0, 8.0, 17.0, 32.0] \)
  
  These values are incorrect for negative \( x \).

### Conclusion:
**Option A** is the correct one, as it most closely matches the computed values.

Let me know if you need further details or explanations!

Here are five follow-up questions you might consider:
1. What is the general behavior of exponential functions like \( 2^x \) as \( x \) becomes negative?
2. How does squaring negative numbers affect the function's output?
3. Can you derive the derivative of \( f(x) = 2^x + x^2 \)?
4. How would \( f(x) \) change if the exponent base was something other than 2?
5. What happens to \( f(x) \) as \( x \to \infty \) or \( x \to -\infty \)?

**Tip**: When comparing function outputs, always be mindful of rounding differences, especially when dealing with irrational numbers or exponential functions.

Learn how to evaluate the function f(x) = 2^x + x^2 for multiple values of x and compare the results with options presented in a table format. This problem involves both exponential and quadratic functions. Suitable for students in Grades 9-12.

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