Let me review the content in the image and provide a detailed response. I'll analyze the calculations and the logic step by step to ensure everything is correct.From your image, you seem to be analyzing the function $h(x) = \frac{x^2 + 4}{x^2 + 1}$ and determining its behavior under certain conditions. Here's a breakdown of what I observed:

Key Steps in Your Work:

1. Range Calculation:

   • You're analyzing the range of $h(x)$, which involves observing the output values for different $x$.
   • You substituted $x = 0$: $h(0) = \frac{0^2 + 4}{0^2 + 1} = \frac{4}{1} = 4.$

2. Behavior of $h(x)$ as $x \to \infty$:

   • You noted that the degree of the numerator ($x^2$) is the same as the degree of the denominator ($x^2$), so the function's behavior is determined by the leading coefficients.
   • As $x \to \infty$, $h(x) \to 1$ because: $h(x) \sim \frac{x^2}{x^2} = 1.$

3. Substitution to Verify Larger Values of $x$:

   • You substituted $x = 20$: $h(20) = \frac{20^2 + 4}{20^2 + 1} = \frac{400 + 4}{400 + 1} = \frac{404}{401}.$ This is approximately $1.007$, confirming that $h(x)$ approaches 1 but remains slightly above it for large $x$.

4. Conclusion for Range:

   • Based on your calculations, $h(x)$ is bounded within $1 < h(x) \leq 4$.

Verification:

1. Substitution at $x = 0$:

   • Correctly computed $h(0) = 4$.

2. Behavior as $x \to \infty$:

   • $h(x) \to 1$ is accurate for rational functions where numerator and denominator have equal degrees.

3. Substitution of $x = 20$:

   • The computation of $h(20) \approx 1.007$ is correct.

4. Range:

   • Your range conclusion of $1 < h(x) \leq 4$ matches the function's behavior.

Final Assessment:

Your solution and calculations are correct! Everything follows logically and is mathematically sound.

Do you want a detailed explanation of any step?

Here are five related questions you might want to explore:

1. Why does $h(x) \to 1$ as $x \to \infty$?
2. How can you formally prove the range $1 < h(x) \leq 4$ using calculus?
3. What happens to $h(x)$ as $x \to -\infty$?
4. Could there be any critical points in $h(x)$?
5. How would $h(x)$ behave if the numerator had a higher degree than the denominator?

Tip: To solidify your understanding, graph $h(x)$ to observe its behavior visually!