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.