explain how to find the range of this function: h(x)=3+$$ \sqrt{-\left\vert x-4\right\vert+7} $$

To find the range of the function $ h(x) = 3 + \sqrt{-|x - 4| + 7} $, we need to follow a step-by-step approach to identify the possible values that $ h(x) $ can take.

### Step 1: Understand the structure of the function
The function consists of two main parts:
- The constant $ 3 $
- The square root expression $ \sqrt{-|x - 4| + 7} $

The square root function $ \sqrt{y} $ is defined only for non-negative values of $ y $, i.e., $ y \geq 0 $. Therefore, for $ h(x) $ to be defined, the expression inside the square root, $ -|x - 4| + 7 $, must be non-negative.

### Step 2: Analyze the domain restriction from the square root
We require:
\[
-|x - 4| + 7 \geq 0
\]
This simplifies to:
\[
-|x - 4| \geq -7
\]
or equivalently:
\[
|x - 4| \leq 7
\]
This inequality tells us that the absolute value of $ x - 4 $ must be less than or equal to 7. Solving for $ x $, we get:
\[
-7 \leq x - 4 \leq 7
\]
Adding 4 to all sides:
\[
-3 \leq x \leq 11
\]
Thus, the domain of the function is $ x \in [-3, 11] $.

### Step 3: Express the range of the function
Now, let's determine the values that $ h(x) $ can take by considering the minimum and maximum possible values of the square root expression.

We know from the domain analysis that:
\[
|x - 4| \leq 7
\]
Therefore, $ -|x - 4| $ will range from $ -7 $ to $ 0 $ (since the absolute value is non-negative, and we are multiplying by $ -1 $).

So, $ -|x - 4| + 7 $ will range from:
\[
-7 + 7 = 0 \quad \text{to} \quad 0 + 7 = 7
\]

Hence, the expression inside the square root, $ -|x - 4| + 7 $, will range from $ 0 $ to $ 7 $.

### Step 4: Apply the square root
Since $ \sqrt{y} $ is defined for non-negative $ y $, the range of the square root $ \sqrt{-|x - 4| + 7} $ will be from $ \sqrt{0} = 0 $ to $ \sqrt{7} $.

### Step 5: Add the constant
Finally, adding the constant $ 3 $ to the square root, we get the range of $ h(x) $:
\[
h(x) = 3 + \sqrt{-|x - 4| + 7}
\]
Since $ \sqrt{-|x - 4| + 7} $ ranges from $ 0 $ to $ \sqrt{7} $, the range of $ h(x) $ is:
\[
h(x) \in [3, 3 + \sqrt{7}]
\]

Thus, the range of the function $ h(x) $ is:
\[
\boxed{[3, 3 + \sqrt{7}]}
\]

### Do you want further details on this solution, or have any questions?

Here are 5 related questions to deepen understanding:
1. What is the domain of the function $ f(x) = \sqrt{-|x - 5| + 10} $?
2. How would the range of the function change if the constant added was 5 instead of 3?
3. What happens to the range if $ h(x) $ is multiplied by 2 instead of adding 3?
4. How can you graph the function $ h(x) = 3 + \sqrt{-|x - 4| + 7} $?
5. How would the function behave if the square root were removed from the expression?

**Tip:** Always start by identifying any restrictions from the square root or other operations when finding the range.

Explain how to find the range of this function: h(x)=3+$$ \sqrt{-\left\vert x-4\right\vert+7} $$

Explore how to find the range of the function h(x) = 3 + √(-|x - 4| + 7). This problem explains key mathematical concepts like absolute value, square root, and function range. Detailed solution suitable for students in Grades 10-12.

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