Write the domain of the function in interval notation. $$ k(u)=\sqrt{\frac{4 u}{u+7}} $$

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We start with the function $$ k(u)=\sqrt{\frac{4u}{u+7}}. $$ Since square roots require nonnegative arguments, the expression inside the root must satisfy $$ \frac{4u}{u+7} \geq 0. $$ Additionally, the denominator cannot be zero, so $$ u+7 \neq 0 \quad \Longrightarrow \quad u \neq -7. $$ **Step 1. Identify critical points:** The points where the numerator $$4u$$ or the denominator $$u+7$$ are zero are: - $$4u=0 \quad \Longrightarrow \quad u=0$$, - $$u+7=0 \quad \Longrightarrow \quad u=-7$$. **Step 2. Analyze the sign of $$\frac{4u}{u+7}$$ in the different intervals:** We have three intervals to consider: 1. $$u < -7$$ 2. $$-7 < u < 0$$ 3. $$u > 0$$ - **For $$u < -7$$:** - Numerator: $$4u$$ is negative (since $$u$$ is negative). - Denominator: $$u+7$$ is negative (since $$u$$ is less than $$-7$$). - A negative divided by a negative is positive, so $$\frac{4u}{u+7} > 0$$. - **For $$-7 < u < 0$$:** - Numerator: $$4u$$ is negative. - Denominator: $$u+7$$ is positive (since $$u > -7$$). - A negative divided by a positive is negative, so $$\frac{4u}{u+7} < 0$$. - **For $$u > 0$$:** - Numerator: $$4u$$ is positive. - Denominator: $$u+7$$ is positive. - A positive divided by a positive is positive, so $$\frac{4u}{u+7} > 0$$. **Step 3. Check the endpoints:** - At $$u=0$$, the expression becomes $$\frac{4(0)}{0+7}=0$$, which is allowed. - $$u=-7$$ is excluded because it makes the denominator zero. **Conclusion:** The expression under the square root is nonnegative for $$ u < -7 \quad \text{and} \quad u \geq 0. $$ Thus, the domain of $$k(u)$$ in interval notation is $$ (-\infty, -7) \cup [0, \infty). $$ The domain of the function $$ k(u) = \sqrt{\frac{4u}{u+7}} $$ is all real numbers less than -7 and all real numbers greater than or equal to 0. In interval notation, this is written as $$ (-\infty, -7) \cup [0, \infty) $$.

Write the domain of the function in interval notation.

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