determine whether the following functions are Even, Odd, or Neither.
f(x)= 3 square root of x

To determine whether the function \( f(x) = 3\sqrt{x} \) is even, odd, or neither, we need to test it against the definitions of even and odd functions.

### Definitions:
- **Even function**: A function is even if \( f(-x) = f(x) \) for all \( x \) in the domain.
- **Odd function**: A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain.

### Step 1: Check the domain of the function.
The function \( f(x) = 3\sqrt{x} \) is defined for \( x \geq 0 \) (since the square root function is only defined for non-negative values of \( x \)).

### Step 2: Calculate \( f(-x) \).
Let's substitute \( -x \) into the function:

\[
f(-x) = 3\sqrt{-x}
\]

However, \( \sqrt{-x} \) is not defined for \( x \geq 0 \), which means \( f(-x) \) is undefined in the domain of \( f(x) \).

### Conclusion:
Since \( f(-x) \) is not defined for any \( x \) in the domain of \( f(x) \), the function cannot be classified as even or odd. Therefore, the function \( f(x) = 3\sqrt{x} \) is **neither even nor odd**.

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the domain of the function \( f(x) = \sqrt{x} \)?
2. How can we determine if a function is even if its domain is not symmetric around the origin?
3. Is the function \( g(x) = x^2 \) even, odd, or neither?
4. What is the difference between even and odd functions in terms of symmetry?
5. Can a function be both even and odd? If so, what kind of function would that be?

**Tip:** Always check the domain of a function first before determining if it is even, odd, or neither.

determine whether the following functions are Even, Odd, or Neither. f(x)= 3 square root of x

Explore the process of determining whether the function f(x) = 3√x is even, odd, or neither. This problem involves understanding the definitions of even and odd functions and analyzing the domain of the square root function. Suitable for students in Grades 10-12.

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