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

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