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Answer :
- The function is $h(x) = \sqrt{x-7} + 5$.
- The expression inside the square root must be non-negative: $x-7 \geq 0$.
- Solve the inequality: $x \geq 7$.
- The domain of the function is $x \geq 7$, so the answer is $\boxed{x \geq 7}$.
### Explanation
1. Understanding the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have the function $h(x) = \sqrt{x-7} + 5$. The square root function is only defined for non-negative values. Therefore, the expression inside the square root must be greater than or equal to zero.
2. Solving the Inequality
We need to find the values of $x$ for which $x-7 \geq 0$. To solve this inequality, we add 7 to both sides:$$x - 7 + 7 \geq 0 + 7$$$$x \geq 7$$
3. Determining the Domain
The domain of the function $h(x)$ is all $x$ values greater than or equal to 7. This means that $x$ can be 7, or any number larger than 7.
4. Final Answer
The correct answer is C. $x \geq 7$.
### Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if you're modeling the growth of a plant where $x$ represents time in days and $h(x)$ represents the height of the plant, the domain tells you the valid time intervals for which the model makes sense. You can't have negative time, and in this case, the plant model only starts being valid after 7 days. This ensures that your predictions are based on realistic and meaningful data.
- The expression inside the square root must be non-negative: $x-7 \geq 0$.
- Solve the inequality: $x \geq 7$.
- The domain of the function is $x \geq 7$, so the answer is $\boxed{x \geq 7}$.
### Explanation
1. Understanding the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have the function $h(x) = \sqrt{x-7} + 5$. The square root function is only defined for non-negative values. Therefore, the expression inside the square root must be greater than or equal to zero.
2. Solving the Inequality
We need to find the values of $x$ for which $x-7 \geq 0$. To solve this inequality, we add 7 to both sides:$$x - 7 + 7 \geq 0 + 7$$$$x \geq 7$$
3. Determining the Domain
The domain of the function $h(x)$ is all $x$ values greater than or equal to 7. This means that $x$ can be 7, or any number larger than 7.
4. Final Answer
The correct answer is C. $x \geq 7$.
### Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if you're modeling the growth of a plant where $x$ represents time in days and $h(x)$ represents the height of the plant, the domain tells you the valid time intervals for which the model makes sense. You can't have negative time, and in this case, the plant model only starts being valid after 7 days. This ensures that your predictions are based on realistic and meaningful data.
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