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The task is to find the **maximal domain** and the **range** of the following functions, expressed as compositions of simpler functions:

### (a) \( f(x) = \frac{x^2}{x^2 + 1} \)

#### Domain:
This function is a rational function. Since the denominator \(x^2 + 1\) is always positive for all real \(x\), there are no restrictions on \(x\). Therefore, the domain is:
\[
D(f) = \mathbb{R}
\]
(all real numbers).

#### Range:
Let's analyze the behavior of \( f(x) = \frac{x^2}{x^2 + 1} \):

- As \( x \to \infty \), \( f(x) \to 1 \).
- As \( x \to 0 \), \( f(0) = 0 \).
- \( f(x) \) is symmetric about the y-axis (even function), and its value increases as \( |x| \) increases from 0 to infinity.

Since \( 0 \leq f(x) < 1 \) for all \(x \in \mathbb{R}\), the range is:
\[
R(f) = [0, 1)
\]

---

### (b) \( f(x) = \sqrt{x(4 - x)} \)

#### Domain:
To ensure the function is real, the expression under the square root must be non-negative:
\[
x(4 - x) \geq 0
\]
This is a quadratic inequality. The roots of the equation \( x(4 - x) = 0 \) are \(x = 0\) and \(x = 4\). The quadratic opens downwards, meaning \(x(4 - x) \geq 0\) for \( 0 \leq x \leq 4 \). Therefore, the domain is:
\[
D(f) = [0, 4]
\]

#### Range:
The expression inside the square root is a parabola with its maximum at \(x = 2\), where:
\[
f(2) = \sqrt{2(4 - 2)} = \sqrt{4} = 2
\]
- At \(x = 0\), \(f(0) = \sqrt{0} = 0\).
- At \(x = 4\), \(f(4) = \sqrt{0} = 0\).

Thus, the range of the function is:
\[
R(f) = [0, 2]
\]

---

Would you like further details on these solutions, or have any specific questions?

### Related Questions:
1. How would the domain and range change if the function in part (a) were modified to \( \frac{x^3}{x^2 + 1} \)?
2. Can you graph the function \( f(x) = \frac{x^2}{x^2 + 1} \) and describe its critical points?
3. What is the effect of adding a constant inside the square root in part (b), like \( \sqrt{x(4 - x) + 1} \)?
4. How can you find the inverse function for \( f(x) = \frac{x^2}{x^2 + 1} \), if it exists?
5. If you shift the quadratic inside the square root to \( \sqrt{x(5 - x)} \), how does that affect the domain and range?

### Tip:
Always check for restrictions in the domain by analyzing the denominator (in rational functions) or expressions under square roots and logarithms.

Найти максимальную область определения и множество значений следующих функций (представляя функции в виде композиции простых): (a) f(x) = x^2 / (x^2 + 1); (b) f(x) = √(x(4 - x)).

Find the maximal domain and range of two functions: f(x) = x^2 / (x^2 + 1) and f(x) = √(x(4 - x)). This problem involves analyzing rational functions and quadratic inequalities, suitable for students in Grades 9-12.

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