The domain of the function \(\mathrm{f(x)} = \sqrt{\sqrt{\mathrm{x} + 2} - \sqrt{4 - \mathrm{x}}}\) is the set of real numbers...

1. Translate the problem requirements: Identify what "domain" means - the set of x values where \(\mathrm{f(x) = \sqrt{\sqrt{x + 2} - \sqrt{4 - x}}}\) is defined, which requires all expressions under square roots to be non-negative
2. Establish basic constraints: Determine the fundamental restrictions from the inner square roots \(\sqrt{\mathrm{x + 2}}\) and \(\sqrt{\mathrm{4 - x}}\)
3. Analyze the critical constraint: Determine when the expression under the outer square root, \(\sqrt{\mathrm{x + 2}} - \sqrt{\mathrm{4 - x}}\), is non-negative
4. Solve the inequality: Find where \(\sqrt{\mathrm{x + 2}} \geq \sqrt{\mathrm{4 - x}}\) by squaring both sides and solving the resulting inequality
5. Combine all restrictions: Intersect all constraints to find the final domain

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. The "domain" of a function is simply the set of all x-values where the function actually works - where we can plug in x and get a real number out.

For our function \(\mathrm{f(x) = \sqrt{\sqrt{x + 2} - \sqrt{4 - x}}}\), we need to think about when square roots cause problems. A square root is only defined for non-negative numbers (zero or positive). So we need:

• Everything under every square root to be non-negative
• This includes the inner square roots \(\sqrt{\mathrm{x + 2}}\) and \(\sqrt{\mathrm{4 - x}}\)
• And crucially, the expression under the outer square root: \(\sqrt{\mathrm{x + 2}} - \sqrt{\mathrm{4 - x}}\)

Process Skill: TRANSLATE - Converting the domain question into concrete mathematical requirements

2. Establish basic constraints

Let's find the basic restrictions from the inner square roots first.

For \(\sqrt{\mathrm{x + 2}}\) to be defined: \(\mathrm{x + 2 \geq 0}\), which means \(\mathrm{x \geq -2}\)
For \(\sqrt{\mathrm{4 - x}}\) to be defined: \(\mathrm{4 - x \geq 0}\), which means \(\mathrm{x \leq 4}\)

So far, we know x must satisfy both conditions: \(\mathrm{-2 \leq x \leq 4}\)

But we're not done yet! We also need the outer square root to work.

3. Analyze the critical constraint

The trickiest part is making sure the expression under the outer square root is non-negative:
\(\sqrt{\mathrm{x + 2}} - \sqrt{\mathrm{4 - x}} \geq 0\)

In plain English: the first square root must be at least as big as the second square root.
This means: \(\sqrt{\mathrm{x + 2}} \geq \sqrt{\mathrm{4 - x}}\)

Process Skill: INFER - Recognizing that the outer square root creates an additional constraint

4. Solve the inequality

To solve \(\sqrt{\mathrm{x + 2}} \geq \sqrt{\mathrm{4 - x}}\), we can square both sides (since both sides are non-negative in our valid range):

Squaring both sides: \(\mathrm{x + 2 \geq 4 - x}\)
Adding x to both sides: \(\mathrm{2x + 2 \geq 4}\)
Subtracting 2: \(\mathrm{2x \geq 2}\)
Dividing by 2: \(\mathrm{x \geq 1}\)

So we need \(\mathrm{x \geq 1}\) for the outer square root to work.

5. Combine all restrictions

Now we combine all our requirements:

• From \(\sqrt{\mathrm{x + 2}}\): \(\mathrm{x \geq -2}\)
• From \(\sqrt{\mathrm{4 - x}}\): \(\mathrm{x \leq 4}\)
• From \(\sqrt{\sqrt{\mathrm{x + 2}} - \sqrt{\mathrm{4 - x}}}\): \(\mathrm{x \geq 1}\)

The intersection of these constraints is: \(\mathrm{1 \leq x \leq 4}\)

Process Skill: APPLY CONSTRAINTS - Combining multiple restrictions to find the final domain

4. Final Answer

The domain is \(\mathrm{1 \leq x \leq 4}\), which matches answer choice D.

To verify: at \(\mathrm{x = 1}\), we get \(\sqrt{\sqrt{3} - \sqrt{3}} = \sqrt{0} = 0\) ✓
At \(\mathrm{x = 4}\), we get \(\sqrt{\sqrt{6} - \sqrt{0}} = \sqrt{6}\) ✓
For any \(\mathrm{x < 1}\), the expression under the outer square root becomes negative, making the function undefined.

Common Faltering Points

Errors while devising the approach

Many students identify the constraints from the inner square roots (\(\mathrm{x \geq -2}\) and \(\mathrm{x \leq 4}\)) but completely forget that the expression under the outer square root must also be non-negative. They conclude the domain is \(\mathrm{-2 \leq x \leq 4}\), missing the critical requirement that \(\sqrt{\mathrm{x + 2}} - \sqrt{\mathrm{4 - x}} \geq 0\).

Students often struggle with the concept that in nested functions like \(\sqrt{\sqrt{\mathrm{x + 2}} - \sqrt{\mathrm{4 - x}}}\), ALL layers must be defined simultaneously. They might think they only need to check one layer of square roots rather than recognizing this is a composition of functions where each component adds its own constraints.

Errors while executing the approach

When solving \(\sqrt{\mathrm{x + 2}} \geq \sqrt{\mathrm{4 - x}}\), students may square both sides without ensuring both sides are non-negative first, or they might flip the inequality sign incorrectly during the squaring process, leading to wrong algebraic results.

After correctly identifying all three constraints (\(\mathrm{x \geq -2}\), \(\mathrm{x \leq 4}\), and \(\mathrm{x \geq 1}\)), students often make errors when finding their intersection. They might incorrectly combine them as \(\mathrm{-2 \leq x \leq 1}\) instead of the correct \(\mathrm{1 \leq x \leq 4}\), confusing which constraint is most restrictive.

Errors while selecting the answer

Students who correctly identify some but not all constraints might select answer choice B (\(\mathrm{-2 \leq x \leq 4}\)) based only on the inner square root constraints, or answer choice A (\(\mathrm{-2 \leq x \leq 1}\)) if they incorrectly combine the constraints, rather than the complete domain \(\mathrm{1 \leq x \leq 4}\).