Given that \( f \) is the function detine by \( f(x)=\sqrt{\frac{2-x}{3+x}} \) Determine whe the functian \( f(x) \) is continuous ano whether the is a left or. right continuen of the interval of Continuity

To determine the continuity of the function \( f(x) = \sqrt{\frac{2-x}{3+x}} \), we need to analyze the expression under the square root and the domain of the function. 1. **Domain of the function**: The function \( f(x) \) is defined when the expression inside the square root is non-negative, and the denominator is not zero. Therefore, we need to solve the following inequalities: - The expression \( \frac{2-x}{3+x} \) must be non-negative: \[ \frac{2-x}{3+x} \geq 0 \] - The denominator \( 3+x \) must not be zero: \[ 3+x \neq 0 \implies x \neq -3 \] 2. **Finding the intervals where \( \frac{2-x}{3+x} \geq 0 \)**: - The critical points occur when the numerator \( 2-x = 0 \) and the denominator \( 3+x = 0 \): - \( 2-x = 0 \) gives \( x = 2 \) - \( 3+x = 0 \) gives \( x = -3 \) - We will test the sign of \( \frac{2-x}{3+x} \) in the intervals determined by these critical points: \( (-\infty, -3) \), \( (-3, 2) \), and \( (2, \infty) \). - **Interval \( (-\infty, -3) \)**: - Choose \( x = -4 \): \[ \frac{2 - (-4)}{3 + (-4)} = \frac{6}{-1} < 0 \] - **Interval \( (-3, 2) \)**: - Choose \( x = 0 \): \[ \frac{2 - 0}{3 + 0} = \frac{2}{3} > 0 \] - **Interval \( (2, \infty) \)**: - Choose \( x = 3 \): \[ \frac{2 - 3}{3 + 3} = \frac{-1}{6} < 0 \] 3. **Summary of signs**: - \( (-\infty, -3) \): Negative - \( (-3, 2) \): Positive - \( (2, \infty) \): Negative 4. **Determining the intervals of continuity**: - The function \( f(x) \) is defined and continuous in the interval \( (-3, 2) \) because: - The expression \( \frac{2-x}{3+x} \) is non-negative in this interval. - The function is not defined at \( x = -3 \) and \( x = 2 \). 5. **Left and right continuity**: - At \( x = -3 \): The function is not defined, so it is not continuous. - At \( x = 2 \): The function is not defined, so it is not continuous. Thus, the function \( f(x) \) is continuous on the interval \( (-3, 2) \). The left endpoint \( -3 \) is not included, and the right endpoint \( 2 \) is also not included. Therefore, the interval of continuity is \( (-3, 2) \).