Answer :
- The domain of $h(x) = \sqrt{x-7} + 5$ is determined by the requirement that the expression inside the square root must be non-negative.
- Set up the inequality $x - 7 \geq 0$.
- Solve the inequality to find $x \geq 7$.
- The domain of the function is $x \geq 7$, so the answer is $\boxed{x \geq 7}$.
### Explanation
1. Understanding the Problem
We are asked to find the domain of the function $h(x) =
\sqrt{x-7} + 5$. 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 a square root function, and we know that the expression inside the square root must be non-negative.
2. Setting up the Inequality
To find the domain, we need to solve the inequality $x - 7
\geq 0$.
3. Solving the Inequality
Adding 7 to both sides of the inequality, we get $x
\geq 7$.
4. Determining the Domain
This means that the domain of the function $h(x)$ is all real numbers $x$ such that $x$ is greater than or equal to 7.
5. Selecting the Correct Answer
Comparing our solution to the given options, we see that option D, $x \geq 7$, is the correct answer.
### Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if $h(x)$ represents the height of a plant after $x$ days since planting, then the domain $x \geq 7$ means we can only start measuring the height of the plant from the 7th day onwards. Before that, the function is not defined in this context. Similarly, in physics, if a function describes the velocity of an object, the domain specifies the time interval for which the velocity is valid.
- Set up the inequality $x - 7 \geq 0$.
- Solve the inequality to find $x \geq 7$.
- The domain of the function is $x \geq 7$, so the answer is $\boxed{x \geq 7}$.
### Explanation
1. Understanding the Problem
We are asked to find the domain of the function $h(x) =
\sqrt{x-7} + 5$. 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 a square root function, and we know that the expression inside the square root must be non-negative.
2. Setting up the Inequality
To find the domain, we need to solve the inequality $x - 7
\geq 0$.
3. Solving the Inequality
Adding 7 to both sides of the inequality, we get $x
\geq 7$.
4. Determining the Domain
This means that the domain of the function $h(x)$ is all real numbers $x$ such that $x$ is greater than or equal to 7.
5. Selecting the Correct Answer
Comparing our solution to the given options, we see that option D, $x \geq 7$, is the correct answer.
### Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if $h(x)$ represents the height of a plant after $x$ days since planting, then the domain $x \geq 7$ means we can only start measuring the height of the plant from the 7th day onwards. Before that, the function is not defined in this context. Similarly, in physics, if a function describes the velocity of an object, the domain specifies the time interval for which the velocity is valid.
