Functions

To determine where the function is monotonically increasing, we need to compute the derivative of the function . Given: First, calculate the derivative: 1. The derivative of is 2. 2. The derivative of is . 3. The derivative of is found using the chain rule: Simplifying: Thus, the derivative of is: Now, for to be monotonically increasing, we need . The derivative simplifies to: After analyzing this expression, we find that the function is monotonically increasing for all real values of . Therefore, the correct answer is , or option (3).

We are given the function . We need to find the domain of the function, i.e., the set of all possible values of for which the function is defined.
1. For the logarithmic term , the argument of the logarithm must be positive, so: This implies that must be less than 3.
2. For the square root term , the argument of the square root must be non-negative, so: This implies that must be greater than or equal to -7. Thus, the domain of the function is restricted to .
3. Additionally, the logarithmic term has a restriction that it cannot be zero because division by zero is undefined. Therefore, we must ensure that: Hence, must be excluded from the domain. So, the domain of the function is: Thus, the correct answer is (C).