If $$ f(x)=\frac{\sqrt{x}+2}{3-3 x^{2}} $$, for which values of $$ x $$ is 1.2.2 1.2.3 1.2.4 1. non real. (x) undefined

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To determine the values of $$ x $$ for which the function
$$
f(x) = \frac{\sqrt{x} + 2}{3 - 3x^2}
$$
is **non-real** or **undefined**, we'll analyze both the numerator and the denominator.

### 1. When is $$ f(x) $$ **non-real**?

For $$ f(x) $$ to yield real numbers:

- **Square Root Constraint**: The expression under the square root must be non-negative.
  $$
  \sqrt{x} \text{ is real} \quad \Rightarrow \quad x \geq 0
  $$

If $$ x < 0 $$, $$ \sqrt{x} $$ becomes non-real (a complex number), making the entire function $$ f(x) $$ non-real.

**Conclusion**:
$$
f(x) \text{ is non-real for all } x < 0.
$$

### 2. When is $$ f(x) $$ **undefined**?

The function $$ f(x) $$ is undefined when the denominator equals zero because division by zero is undefined.

- **Denominator Constraint**:
  $$
  3 - 3x^2 = 0 \quad \Rightarrow \quad 3x^2 = 3 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm 1
  $$

**However**, considering the domain from the square root constraint ($$ x \geq 0 $$):
- $$ x = -1 $$ is already excluded since $$ x < 0 $$ makes $$ f(x) $$ non-real.
- $$ x = 1 $$ is within the allowable domain but makes the denominator zero.

**Conclusion**:
$$
f(x) \text{ is undefined at } x = 1.
$$

### **Summary**

- **Non-real**: $$ f(x) $$ is non-real for all real numbers $$ x $$ where $$ x < 0 $$.
- **Undefined**: $$ f(x) $$ is undefined at $$ x = 1 $$.

### **Graphical Illustration**

To visualize these constraints:

- **Domain for Real Values**: $$ x \in [0, 1) \cup (1, \infty) $$
- **Excluded Points**: $$ x = 1 $$ (makes denominator zero) and $$ x < 0 $$ (makes numerator non-real).

---

**Note**: If considering complex numbers, the function can be extended beyond these constraints, but within the realm of real numbers, the above conditions apply.

**Answer Summary**:
All real numbers x below zero make f (x ) non-real, and x = 1 and x = –1 make f (x ) undefined. In other words:
• f(x) is non-real for every real x with x < 0
• f(x) is undefined at x = 1 and x = −1
$$ f(x) $$ is non-real for all real numbers $$ x $$ where $$ x < 0 $$ and is undefined at $$ x = 1 $$ and $$ x = -1 $$.

If \( f(x)=\frac{\sqrt{x}+2}{3-3 x^{2}} \), for which values of \( x \) is 1.2.2 1.2.3 1.2.4 1. non real. (x) undefined

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