Collection of 26 mathematical equations, each labeled with a letter from a to z, showcasing various algebraic and transcendental functions.

Problem Analysis

The task appears to involve analyzing a collection of functions and determining their properties, such as whether they are even, odd, or neither. The functions are listed in three columns (a-z), and we need to classify each function based on its symmetry properties.

#### Definitions:
1. Even Function: A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \).
2. Odd Function: A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain of \( f \).
3. Neither Even nor Odd: A function that does not satisfy either of the above conditions.

Solution Approach

For each function, we will:
1. Substitute \( -x \) into the function.
2. Simplify the expression to see if it matches \( f(x) \) (even), \( -f(x) \) (odd), or neither.

Let's analyze each function step by step.

---

Column 1: Functions \( a \) to \( i \)

#### a) \( y = x^2 + 2x + 1 \)
- Substitute \( -x \):
\[
f(-x) = (-x)^2 + 2(-x) + 1 = x^2 - 2x + 1
\]
- Compare with \( f(x) = x^2 + 2x + 1 \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### b) \( y = 4x^3 - 3x^2 + 2x - 1 \)
- Substitute \( -x \):
\[
f(-x) = 4(-x)^3 - 3(-x)^2 + 2(-x) - 1 = -4x^3 - 3x^2 - 2x - 1
\]
- Compare with \( f(x) = 4x^3 - 3x^2 + 2x - 1 \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### c) \( y = \frac{1}{4}x^4 + \frac{1}{3}x^3 + \frac{1}{2}x^2 \)
- Substitute \( -x \):
\[
f(-x) = \frac{1}{4}(-x)^4 + \frac{1}{3}(-x)^3 + \frac{1}{2}(-x)^2 = \frac{1}{4}x^4 - \frac{1}{3}x^3 + \frac{1}{2}x^2
\]
- Compare with \( f(x) = \frac{1}{4}x^4 + \frac{1}{3}x^3 + \frac{1}{2}x^2 \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### d) \( y = -\frac{5x^3}{9} - \frac{8x^7}{13} - \frac{9x^6}{16} \)
- Substitute \( -x \):
\[
f(-x) = -\frac{5(-x)^3}{9} - \frac{8(-x)^7}{13} - \frac{9(-x)^6}{16} = \frac{5x^3}{9} + \frac{8x^7}{13} - \frac{9x^6}{16}
\]
- Compare with \( f(x) = -\frac{5x^3}{9} - \frac{8x^7}{13} - \frac{9x^6}{16} \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### e) \( y = (3x - 5)^3 \)
- Substitute \( -x \):
\[
f(-x) = (3(-x) - 5)^3 = (-3x - 5)^3 = -(3x + 5)^3
\]
- Compare with \( f(x) = (3x - 5)^3 \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### f) \( y = (\sqrt{x} - 1)^2 - (x^2 + 1)^4 \)
- Note: The domain of \( \sqrt{x} \) is \( x \geq 0 \), so this function is not defined for negative \( x \). Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### g) \( y = x^{11} - x^9 + x^7 - x^5 \)
- Substitute \( -x \):
\[
f(-x) = (-x)^{11} - (-x)^9 + (-x)^7 - (-x)^5 = -x^{11} + x^9 - x^7 + x^5 = -(x^{11} - x^9 + x^7 - x^5)
\]
- Compare with \( f(x) = x^{11} - x^9 + x^7 - x^5 \):
\[
f(-x) = -f(x)
\]
- Conclusion: Odd.

#### h) \( y = x^{-5} + x^{-7} + x^{-9} - 11 \)
- Substitute \( -x \):
\[
f(-x) = (-x)^{-5} + (-x)^{-7} + (-x)^{-9} - 11 = -x^{-5} - x^{-7} - x^{-9} - 11
\]
- Compare with \( f(x) = x^{-5} + x^{-7} + x^{-9} - 11 \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### i) \( y = \frac{8}{x^8} - \frac{6}{x^6} + \frac{4}{x^4} - \frac{2}{x^2} \)
- Substitute \( -x \):
\[
f(-x) = \frac{8}{(-x)^8} - \frac{6}{(-x)^6} + \frac{4}{(-x)^4} - \frac{2}{(-x)^2} = \frac{8}{x^8} - \frac{6}{x^6} + \frac{4}{x^4} - \frac{2}{x^2}
\]
- Compare with \( f(x) = \frac{8}{x^8} - \frac{6}{x^6} + \frac{4}{x^4} - \frac{2}{x^2} \):
\[
f(-x) = f(x)
\]
- Conclusion: Even.

---

Column 2: Functions \( j \) to \( r \)

#### j) \( y = x + \sqrt{x} + \sqrt[3]{x} + \sqrt[5]{x} \)
- Note: The domain of \( \sqrt{x} \) is \( x \geq 0 \), so this function is not defined for negative \( x \). Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### k) \( y = \sqrt[3]{x^8} - \sqrt[4]{x^7} + \sqrt[5]{x^6} \)
- Substitute \( -x \):
\[
f(-x) = \sqrt[3]{(-x)^8} - \sqrt[4]{(-x)^7} + \sqrt[5]{(-x)^6} = \sqrt[3]{x^8} - \sqrt[4]{-x^7} + \sqrt[5]{x^6}
\]
- Compare with \( f(x) = \sqrt[3]{x^8} - \sqrt[4]{x^7} + \sqrt[5]{x^6} \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### l) \( y = \frac{1}{\sqrt[3]{2x^4}} - \frac{1}{\sqrt[4]{2x^3}} \)
- Substitute \( -x \):
\[
f(-x) = \frac{1}{\sqrt[3]{2(-x)^4}} - \frac{1}{\sqrt[4]{2(-x)^3}} = \frac{1}{\sqrt[3]{2x^4}} - \frac{1}{\sqrt[4]{-2x^3}}
\]
- Compare with \( f(x) = \frac{1}{\sqrt[3]{2x^4}} - \frac{1}{\sqrt[4]{2x^3}} \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### m) \( y = \frac{\sqrt[5]{4x}}{5} + \sqrt[4]{\frac{1}{x^3}} \)
- Substitute \( -x \):
\[
f(-x) = \frac{\sqrt[5]{4(-x)}}{5} + \sqrt[4]{\frac{1}{(-x)^3}} = \frac{\sqrt[5]{-4x}}{5} + \sqrt[4]{-\frac{1}{x^3}}
\]
- Compare with \( f(x) = \frac{\sqrt[5]{4x}}{5} + \sqrt[4]{\frac{1}{x^3}} \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### n) \( y = \sqrt{x^5} - \sqrt[4]{x^9} \)
- Note: The domain of \( \sqrt{x^5} \) and \( \sqrt[4]{x^9} \) is \( x \geq 0 \), so this function is not defined for negative \( x \). Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### o) \( y = \sqrt{x \sqrt{x}} + \sqrt[3]{x^2 \sqrt{x^3}} \)
- Note: The domain of \( \sqrt{x} \) and \( \sqrt[3]{x} \) is \( x \geq 0 \), so this function is not defined for negative \( x \). Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### p) \( y = \sqrt{x^3 \sqrt{x^5 \sqrt{x^7}}} \)
- Note: The domain of \( \sqrt{x} \) is \( x \geq 0 \), so this function is not defined for negative \( x \). Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### q) \( y = \frac{5x^{-3} \cdot \sqrt{x^4} \cdot \sqrt[3]{x^5}}{8x^9 \cdot \sqrt[5]{x^4} \cdot \sqrt[7]{x^{11}} \cdot \sqrt[9]{x}} \)
- Substitute \( -x \):
\[
f(-x) = \frac{5(-x)^{-3} \cdot \sqrt{(-x)^4} \cdot \sqrt[3]{(-x)^5}}{8(-x)^9 \cdot \sqrt[5]{(-x)^4} \cdot \sqrt[7]{(-x)^{11}} \cdot \sqrt[9]{-x}}
\]
- Simplify each term:
\[
(-x)^{-3} = -x^{-3}, \quad \sqrt{(-x)^4} = \sqrt{x^4} = x^2, \quad \sqrt[3]{(-x)^5} = -\sqrt[3]{x^5}
\]
\[
(-x)^9 = -x^9, \quad \sqrt[5]{(-x)^4} = \sqrt[5]{x^4}, \quad \sqrt[7]{(-x)^{11}} = -\sqrt[7]{x^{11}}, \quad \sqrt[9]{-x} = -\sqrt[9]{x}
\]
- Combine:
\[
f(-x) = \frac{5(-x^{-3}) \cdot x^2 \cdot (-\sqrt[3]{x^5})}{8(-x^9) \cdot \sqrt[5]{x^4} \cdot (-\sqrt[7]{x^{11}}) \cdot (-\sqrt[9]{x})}
\]
- Simplify:
\[
f(-x) = \frac{-5x^{-3} \cdot x^2 \cdot (-\sqrt[3]{x^5})}{-8x^9 \cdot \sqrt[5]{x^4} \cdot (-\sqrt[7]{x^{11}}) \cdot (-\sqrt[9]{x})}
\]
\[
f(-x) = \frac{5x^{-1} \cdot \sqrt[3]{x^5}}{-8x^9 \cdot \sqrt[5]{x^4} \cdot \sqrt[7]{x^{11}} \cdot \sqrt[9]{x}}
\]
- Compare with \( f(x) \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### r) \( y = \sqrt{\frac{\sqrt[4]{x^7}}{6x^3}} + \sqrt[3]{\frac{\sqrt{x^8} \cdot \sqrt{x^9}}{\sqrt[4]{4x^4 \sqrt{x^5}}}} \)
- Substitute \( -x \):
\[
f(-x) = \sqrt{\frac{\sqrt[4]{(-x)^7}}{6(-x)^3}} + \sqrt[3]{\frac{\sqrt{(-x)^8} \cdot \sqrt{(-x)^9}}{\sqrt[4]{4(-x)^4 \sqrt{(-x)^5}}}}
\]
- Simplify each term:
\[
\sqrt[4]{(-x)^7} = \sqrt[4]{-x^7}, \quad (-x)^3 = -x^3, \quad \sqrt{(-x)^8} = \sqrt{x^8} = x^4, \quad \sqrt{(-x)^9} = \sqrt{-x^9}
\]
\[
\sqrt[4]{4(-x)^4} = \sqrt[4]{4x^4}, \quad \sqrt{(-x)^5} = \sqrt{-x^5}
\]
- Combine:
\[
f(-x) = \sqrt{\frac{\sqrt[4]{-x^7}}{6(-x^3)}} + \sqrt[3]{\frac{x^4 \cdot \sqrt{-x^9}}{\sqrt[4]{4x^4 \sqrt{-x^5}}}}
\]
- Compare with \( f(x) \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

---

Column 3: Functions \( s \) to \( Z \)

#### s) \( y = \sin x + \cos x + \tan x \)
- Substitute \( -x \):
\[
f(-x) = \sin(-x) + \cos(-x) + \tan(-x) = -\sin x + \cos x - \tan x
\]
- Compare with \( f(x) = \sin x + \cos x + \tan x \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### t) \( y = \log_2 x - \ln x + \log_5 x \)
- Note: The domain of logarithmic functions is \( x > 0 \), so this function is not defined for negative \( x \). Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### u) \( y = 3 \arcsin x - 2 \arctan x \)
- Substitute \( -x \):
\[
f(-x) = 3 \arcsin(-x) - 2 \arctan(-x) = 3(-\arcsin x) - 2(-\arctan x) = -3 \arcsin x + 2 \arctan x
\]
- Compare with \( f(x) = 3 \arcsin x - 2 \arctan x \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### v) \( y = 5 \cot x + 8 \operatorname{arcsec} x \)
- Note: The domain of \( \cot x \) and \( \operatorname{arcsec} x \) excludes certain values of \( x \), and the function is not symmetric around the origin. Therefore, it cannot be even or odd.
- Conclusion: Neither even nor odd.

#### w) \( y = \operatorname{arccot} x - 2 \cot x \)
- Substitute \( -x \):
\[
f(-x) = \operatorname{arccot}(-x) - 2 \cot(-x) = -\operatorname{arccot}(x) + 2 \cot(x)
\]
- Compare with \( f(x) = \operatorname{arccot} x - 2 \cot x \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### x) \( y = x - \ln x + 1 \)
- Substitute \( -x \):
\[
f(-x) = -x - \ln(-x) + 1
\]
- Note: The natural logarithm \( \ln(-x) \) is not defined for real numbers. Therefore, the function is not even or odd.
- Conclusion: Neither even nor odd.

#### y) \( y = 2^x - 3e^x - 4^x \)
- Substitute \( -x \):
\[
f(-x) = 2^{-x} - 3e^{-x} - 4^{-x}
\]
- Compare with \( f(x) = 2^x - 3e^x - 4^x \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### z) \( y = 5 \times 9^x - 4 \times 5^x + \frac{7^x}{\ln 7} \)
- Substitute \( -x \):
\[
f(-x) = 5 \times 9^{-x} - 4 \times 5^{-x} + \frac{7^{-x}}{\ln 7}
\]
- Compare with \( f(x) = 5 \times 9^x - 4 \times 5^x + \frac{7^x}{\ln 7} \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

#### Z) \( y = -6e^x + 5^x - 5x + \frac{x}{5} \)
- Substitute \( -x \):
\[
f(-x) = -6e^{-x} + 5^{-x} - 5(-x) + \frac{-x}{5} = -6e^{-x} + 5^{-x} + 5x - \frac{x}{5}
\]
- Compare with \( f(x) = -6e^x + 5^x - 5x + \frac{x}{5} \):
\[
f(-x) \neq f(x) \quad \text{and} \quad f(-x) \neq -f(x)
\]
- Conclusion: Neither even nor odd.

---

Final Answer

Summarizing the results for all functions:

\[
\boxed{
\begin{array}{ll}
\text{Even:} & i \\
\text{Odd:} & g \\
\text{Neither:} & \text{All others}
\end{array}
}
\]