Even and Odd Function worksheets: Practice and Master Mathematical ... - Free Printable

It seems like the task involves analyzing or solving problems related to the given functions. However, the problem statement is not explicitly provided in your question. To proceed, I will assume that the task is to analyze the properties of these functions, such as their domains, ranges, symmetry, and behavior. If you have a specific problem or task in mind, please clarify.

Given Functions:

1. \( f(x) = x^4 + 5x^2 \)
2. \( f(x) = |x| + x \)
3. \( f(x) = \frac{x}{x^2 + 2} \)
4. \( f(x) = \frac{1}{x - 2} \)
5. \( f(x) = \frac{x^4 + 3}{x^2 + 1} \)
6. \( f(x) = x^2 \sqrt{x^2 + 1} \)

Analysis of Each Function:

#### 1. \( f(x) = x^4 + 5x^2 \)
- Domain: All real numbers, \( \mathbb{R} \), because it is a polynomial.
- Range: Since \( x^4 \geq 0 \) and \( 5x^2 \geq 0 \) for all \( x \), the function is always non-negative. The minimum value occurs at \( x = 0 \), where \( f(0) = 0 \). As \( x \to \pm\infty \), \( f(x) \to \infty \). Thus, the range is \( [0, \infty) \).
- Symmetry: Even function because \( f(-x) = (-x)^4 + 5(-x)^2 = x^4 + 5x^2 = f(x) \).
- Behavior: The function is increasing for \( x > 0 \) and decreasing for \( x < 0 \).

#### 2. \( f(x) = |x| + x \)
- Domain: All real numbers, \( \mathbb{R} \).
- Range: For \( x \geq 0 \), \( f(x) = x + x = 2x \), so \( f(x) \geq 0 \). For \( x < 0 \), \( f(x) = -x + x = 0 \). Thus, the range is \( [0, \infty) \).
- Symmetry: Not symmetric. It is neither even nor odd.
- Behavior: Constant at 0 for \( x < 0 \) and linearly increasing for \( x \geq 0 \).

#### 3. \( f(x) = \frac{x}{x^2 + 2} \)
- Domain: All real numbers, \( \mathbb{R} \), because the denominator \( x^2 + 2 \neq 0 \) for any \( x \).
- Range: The function is continuous and differentiable. As \( x \to \pm\infty \), \( f(x) \to 0 \). The maximum and minimum values can be found by taking the derivative and setting it to zero. The critical points occur where \( f'(x) = 0 \). After solving, the range is \( \left[ -\frac{1}{2\sqrt{2}}, \frac{1}{2\sqrt{2}} \right] \).
- Symmetry: Odd function because \( f(-x) = \frac{-x}{(-x)^2 + 2} = -\frac{x}{x^2 + 2} = -f(x) \).
- Behavior: The function has a horizontal asymptote at \( y = 0 \).

#### 4. \( f(x) = \frac{1}{x - 2} \)
- Domain: All real numbers except \( x = 2 \), because the denominator cannot be zero. So, the domain is \( \mathbb{R} \setminus \{2\} \).
- Range: All real numbers except 0, because the function can take any value except 0. So, the range is \( \mathbb{R} \setminus \{0\} \).
- Symmetry: Neither even nor odd.
- Behavior: Vertical asymptote at \( x = 2 \) and horizontal asymptote at \( y = 0 \).

#### 5. \( f(x) = \frac{x^4 + 3}{x^2 + 1} \)
- Domain: All real numbers, \( \mathbb{R} \), because the denominator \( x^2 + 1 \neq 0 \) for any \( x \).
- Range: The function is continuous and differentiable. As \( x \to \pm\infty \), \( f(x) \to x^2 \). The minimum value can be found by taking the derivative and setting it to zero. After solving, the minimum value is 3, so the range is \( [3, \infty) \).
- Symmetry: Even function because \( f(-x) = \frac{(-x)^4 + 3}{(-x)^2 + 1} = \frac{x^4 + 3}{x^2 + 1} = f(x) \).
- Behavior: The function is increasing for \( x > 0 \) and decreasing for \( x < 0 \).

#### 6. \( f(x) = x^2 \sqrt{x^2 + 1} \)
- Domain: All real numbers, \( \mathbb{R} \), because the expression under the square root is always non-negative.
- Range: Since \( x^2 \geq 0 \) and \( \sqrt{x^2 + 1} \geq 1 \), the function is always non-negative. The minimum value occurs at \( x = 0 \), where \( f(0) = 0 \). As \( x \to \pm\infty \), \( f(x) \to \infty \). Thus, the range is \( [0, \infty) \).
- Symmetry: Even function because \( f(-x) = (-x)^2 \sqrt{(-x)^2 + 1} = x^2 \sqrt{x^2 + 1} = f(x) \).
- Behavior: The function is increasing for \( x > 0 \) and decreasing for \( x < 0 \).

Final Answer:

The analysis of the functions is summarized above. If you have a specific task or problem related to these functions, please provide more details so I can assist further.

\[
\boxed{\text{Analysis provided above}}
\]