To find the range of the function

$y = 2 + \frac{x^2}{x^2 + 4},$

we analyze the behavior of the function and its possible outputs.

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Step 1: Understand the function

The function consists of two parts:

• $2$: A constant.
• $\frac{x^2}{x^2 + 4}$: A fraction involving $x^2$, which is always non-negative ($x^2 \geq 0$).

Thus, the function $y$ is the sum of 2 and a fraction that lies between 0 and 1.

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Step 2: Analyze the fraction $\frac{x^2}{x^2 + 4}$

1. The denominator $x^2 + 4$ is always positive, as $x^2 \geq 0$.
2. As $x^2$ increases, the fraction approaches 1 but never equals it.

Specifically:

• When $x^2 = 0$, $\frac{x^2}{x^2 + 4} = 0$.
• As $x^2 \to \infty$, $\frac{x^2}{x^2 + 4} \to 1$.

Therefore: $0 \leq \frac{x^2}{x^2 + 4} < 1.$

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Step 3: Determine the range of $y$

Adding 2 to the inequality: $2 \leq y < 3.$

Thus, the range of $y$ is: $[2, 3).$

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Final Answer:

The range of the function is: $[2, 3).$

Would you like a more detailed explanation, or have any questions?

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Related Questions:

1. What is the domain of the function $y = 2 + \frac{x^2}{x^2 + 4}$?
2. How does the value of $y$ change as $x \to 0$ or $x \to \infty$?
3. What happens to the function if the constant 2 is replaced with another number, e.g., 3?
4. Can this function ever reach $y = 3$? Why or why not?
5. How can the range be visualized graphically?

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Tip:

When working with rational functions, check the behavior at extreme values (e.g., $x \to \infty$ or $x \to 0$) to understand their limits and range.