Questions: f(x)=sqrt(x+13)/(x+3)(x-2)

Solution Steps

To analyze the function \( f(x) = \frac{\sqrt{x+13}}{(x+3)(x-2)} \), we need to consider the domain of the function. The domain is determined by the conditions that the expression under the square root must be non-negative and the denominator must not be zero. Therefore, we need to find the values of \( x \) for which \( x+13 \geq 0 \) and \( (x+3)(x-2) \neq 0 \).

To find the domain of the function \( f(x) = \frac{\sqrt{x+13}}{(x+3)(x-2)} \), we first need to ensure that the expression under the square root is non-negative. This gives us the condition: \[ x + 13 \geq 0 \implies x \geq -13 \]

Next, we need to identify the points where the denominator is zero, as these points will not be included in the domain. The denominator is zero when: \[ (x + 3)(x - 2) = 0 \implies x = -3 \quad \text{or} \quad x = 2 \]

Now, we combine the conditions from Steps 1 and 2. The domain must satisfy \( x \geq -13 \) while excluding the points \( x = -3 \) and \( x = 2 \). Thus, the domain can be expressed in interval notation as: \[ [-13, -3) \cup (-3, 2) \cup (2, \infty) \]

Final Answer