Comprehensive list of algebraic functions from a to z, including rational, polynomial, and transcendental expressions.

Problem Analysis

The task appears to involve analyzing a collection of functions to determine their properties, such as domain, range, asymptotes, continuity, differentiability, or other characteristics. Since the problem does not specify a particular task (e.g., finding asymptotes, determining domains, etc.), I will assume the goal is to analyze the domain of each function. This is a common first step in understanding the behavior of functions.

Solution Approach

For each function, we need to identify any restrictions on the variable \( x \) that would make the function undefined. These restrictions typically arise from:
1. Division by zero.
2. Taking the square root of a negative number.
3. Taking the logarithm of a non-positive number.
4. Other mathematical operations that are not defined for certain inputs.

I will analyze each function systematically.

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Detailed Analysis of Each Function

#### a) \( y = \frac{x}{x+4} \)
- The denominator \( x + 4 \) cannot be zero.
- Set \( x + 4 = 0 \Rightarrow x = -4 \).
- Domain: \( x \in \mathbb{R} \setminus \{-4\} \).

#### b) \( y = \frac{1-x^2}{x-2} \)
- The denominator \( x - 2 \) cannot be zero.
- Set \( x - 2 = 0 \Rightarrow x = 2 \).
- Domain: \( x \in \mathbb{R} \setminus \{2\} \).

#### c) \( y = \frac{2x^2}{2x-1} \)
- The denominator \( 2x - 1 \) cannot be zero.
- Set \( 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \).
- Domain: \( x \in \mathbb{R} \setminus \left\{\frac{1}{2}\right\} \).

#### d) \( y = \frac{x^2+1}{x} \)
- The denominator \( x \) cannot be zero.
- Set \( x = 0 \).
- Domain: \( x \in \mathbb{R} \setminus \{0\} \).

#### e) \( y = \frac{2x^2 - 1 + 3x^3}{3 - 2x^2} \)
- The denominator \( 3 - 2x^2 \) cannot be zero.
- Set \( 3 - 2x^2 = 0 \Rightarrow 2x^2 = 3 \Rightarrow x^2 = \frac{3}{2} \Rightarrow x = \pm \sqrt{\frac{3}{2}} \).
- Domain: \( x \in \mathbb{R} \setminus \left\{-\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}}\right\} \).

#### f) \( y = \frac{x^2 - 9}{1 - x} \)
- The denominator \( 1 - x \) cannot be zero.
- Set \( 1 - x = 0 \Rightarrow x = 1 \).
- Domain: \( x \in \mathbb{R} \setminus \{1\} \).

#### g) \( y = \frac{x^2 + 6x}{x + 2} \)
- The denominator \( x + 2 \) cannot be zero.
- Set \( x + 2 = 0 \Rightarrow x = -2 \).
- Domain: \( x \in \mathbb{R} \setminus \{-2\} \).

#### h) \( y = \frac{2x}{x^2 + 1} \)
- The denominator \( x^2 + 1 \) is always positive for all real \( x \) (since \( x^2 \geq 0 \)).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### i) \( y = 3x + \frac{3}{x-2} \)
- The term \( \frac{3}{x-2} \) requires \( x - 2 \neq 0 \).
- Set \( x - 2 = 0 \Rightarrow x = 2 \).
- Domain: \( x \in \mathbb{R} \setminus \{2\} \).

#### j) \( y = \frac{2x^3 - 5x^2 + x - 4}{4 - x^2} \)
- The denominator \( 4 - x^2 \) cannot be zero.
- Factorize: \( 4 - x^2 = (2 - x)(2 + x) \).
- Set \( 4 - x^2 = 0 \Rightarrow x = \pm 2 \).
- Domain: \( x \in \mathbb{R} \setminus \{-2, 2\} \).

#### k) \( y = \frac{x^2 - 3}{x^3 - 1} \)
- The denominator \( x^3 - 1 \) cannot be zero.
- Factorize: \( x^3 - 1 = (x - 1)(x^2 + x + 1) \).
- Set \( x^3 - 1 = 0 \Rightarrow x = 1 \) (since \( x^2 + x + 1 \) has no real roots).
- Domain: \( x \in \mathbb{R} \setminus \{1\} \).

#### l) \( y = \frac{x^3 - 3}{x^3 - 1} \)
- The denominator \( x^3 - 1 \) cannot be zero.
- Factorize: \( x^3 - 1 = (x - 1)(x^2 + x + 1) \).
- Set \( x^3 - 1 = 0 \Rightarrow x = 1 \) (since \( x^2 + x + 1 \) has no real roots).
- Domain: \( x \in \mathbb{R} \setminus \{1\} \).

#### m) \( y = \frac{2x^2 - 5x}{x^2 + 1} \)
- The denominator \( x^2 + 1 \) is always positive for all real \( x \).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### n) \( y = \frac{2x}{1 - 3x} \)
- The denominator \( 1 - 3x \) cannot be zero.
- Set \( 1 - 3x = 0 \Rightarrow x = \frac{1}{3} \).
- Domain: \( x \in \mathbb{R} \setminus \left\{\frac{1}{3}\right\} \).

#### o) \( y = \frac{3x^2 - 2x^3 + 4}{4 - 4x + x^2} \)
- The denominator \( 4 - 4x + x^2 \) cannot be zero.
- Factorize: \( 4 - 4x + x^2 = (x - 2)^2 \).
- Set \( (x - 2)^2 = 0 \Rightarrow x = 2 \).
- Domain: \( x \in \mathbb{R} \setminus \{2\} \).

#### p) \( y = \frac{x - 1}{x^3 - 1} \)
- The denominator \( x^3 - 1 \) cannot be zero.
- Factorize: \( x^3 - 1 = (x - 1)(x^2 + x + 1) \).
- Set \( x^3 - 1 = 0 \Rightarrow x = 1 \) (since \( x^2 + x + 1 \) has no real roots).
- Domain: \( x \in \mathbb{R} \setminus \{1\} \).

#### q) \( y = \frac{2x^2 + x + 1}{8x} \)
- The denominator \( 8x \) cannot be zero.
- Set \( 8x = 0 \Rightarrow x = 0 \).
- Domain: \( x \in \mathbb{R} \setminus \{0\} \).

#### r) \( y = \frac{1 + x - 3x^3}{x^2 + x - 2} \)
- The denominator \( x^2 + x - 2 \) cannot be zero.
- Factorize: \( x^2 + x - 2 = (x + 2)(x - 1) \).
- Set \( x^2 + x - 2 = 0 \Rightarrow x = -2 \) or \( x = 1 \).
- Domain: \( x \in \mathbb{R} \setminus \{-2, 1\} \).

#### s) \( y = \frac{2x^4 - 1}{x^4 + 1} \)
- The denominator \( x^4 + 1 \) is always positive for all real \( x \).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### t) \( y = 2x - 3\sqrt[3]{x^2} \)
- The cube root \( \sqrt[3]{x^2} \) is defined for all real \( x \).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### u) \( y = \frac{1}{\sqrt{1 - \ln^2 x}} \)
- The expression inside the square root, \( 1 - \ln^2 x \), must be positive.
- Also, \( \ln x \) requires \( x > 0 \).
- Set \( 1 - \ln^2 x > 0 \Rightarrow \ln^2 x < 1 \Rightarrow -1 < \ln x < 1 \).
- Convert to \( x \): \( e^{-1} < x < e^1 \).
- Domain: \( x \in (e^{-1}, e) \).

#### v) \( y = \frac{x}{\ln x} \)
- The denominator \( \ln x \) cannot be zero, and \( \ln x \) requires \( x > 0 \).
- Set \( \ln x = 0 \Rightarrow x = 1 \).
- Domain: \( x \in (0, 1) \cup (1, \infty) \).

#### w) \( y = x \operatorname{arccot} x \)
- The arccotangent function \( \operatorname{arccot} x \) is defined for all real \( x \).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### x) \( y = xe^{1/x^2} \)
- The exponent \( \frac{1}{x^2} \) requires \( x \neq 0 \).
- Domain: \( x \in \mathbb{R} \setminus \{0\} \).

#### y) \( y = 4xe^{-x^2} \)
- The exponential function \( e^{-x^2} \) is defined for all real \( x \).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### z) \( y = x + e^{-x} \)
- The exponential function \( e^{-x} \) is defined for all real \( x \).
- No restrictions on \( x \).
- Domain: \( x \in \mathbb{R} \).

#### Z) \( y = x + \frac{\ln x}{x} \)
- The term \( \ln x \) requires \( x > 0 \).
- The denominator \( x \) cannot be zero.
- Domain: \( x \in (0, \infty) \).

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Final Answer

The domains of the functions are summarized as follows:

\[
\boxed{
\begin{array}{ll}
\text{a)} & x \in \mathbb{R} \setminus \{-4\} \\
\text{b)} & x \in \mathbb{R} \setminus \{2\} \\
\text{c)} & x \in \mathbb{R} \setminus \left\{\frac{1}{2}\right\} \\
\text{d)} & x \in \mathbb{R} \setminus \{0\} \\
\text{e)} & x \in \mathbb{R} \setminus \left\{-\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}}\right\} \\
\text{f)} & x \in \mathbb{R} \setminus \{1\} \\
\text{g)} & x \in \mathbb{R} \setminus \{-2\} \\
\text{h)} & x \in \mathbb{R} \\
\text{i)} & x \in \mathbb{R} \setminus \{2\} \\
\text{j)} & x \in \mathbb{R} \setminus \{-2, 2\} \\
\text{k)} & x \in \mathbb{R} \setminus \{1\} \\
\text{l)} & x \in \mathbb{R} \setminus \{1\} \\
\text{m)} & x \in \mathbb{R} \\
\text{n)} & x \in \mathbb{R} \setminus \left\{\frac{1}{3}\right\} \\
\text{o)} & x \in \mathbb{R} \setminus \{2\} \\
\text{p)} & x \in \mathbb{R} \setminus \{1\} \\
\text{q)} & x \in \mathbb{R} \setminus \{0\} \\
\text{r)} & x \in \mathbb{R} \setminus \{-2, 1\} \\
\text{s)} & x \in \mathbb{R} \\
\text{t)} & x \in \mathbb{R} \\
\text{u)} & x \in (e^{-1}, e) \\
\text{v)} & x \in (0, 1) \cup (1, \infty) \\
\text{w)} & x \in \mathbb{R} \\
\text{x)} & x \in \mathbb{R} \setminus \{0\} \\
\text{y)} & x \in \mathbb{R} \\
\text{z)} & x \in \mathbb{R} \\
\text{Z)} & x \in (0, \infty) \\
\end{array}
}
\]