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The given function is:

\[
f(x) = \sqrt{2 - e^x}
\]

### Step 1: Identify the domain of the square root function

The square root function is only defined for non-negative values, meaning the expression inside the square root must be greater than or equal to zero:

\[
2 - e^x \geq 0
\]

### Step 2: Solve the inequality

\[
e^x \leq 2
\]

Now, take the natural logarithm of both sides:

\[
x \leq \ln(2)
\]

The value of \(\ln(2)\) is approximately \(0.6931\).

### Step 3: Domain of the function

Thus, the function is continuous for \(x\) values where \(x \leq \ln(2)\). This gives the interval:

\[
(-\infty, \ln(2)]
\]

So, the function \(f(x)\) is continuous on the interval \((- \infty, \ln(2)]\).

Would you like more details or further clarifications on this? 

Here are some related questions for deeper understanding:

1. What happens if the value inside the square root becomes negative?
2. How would the domain change if we had \(f(x) = \sqrt{3 - e^x}\)?
3. Can you determine the range of \(f(x)\)?
4. How does the function behave as \(x\) approaches \(\ln(2)\)?
5. What would the graph of \(f(x)\) look like for this domain?

**Tip:** Whenever dealing with square roots, always ensure that the radicand (the part inside the root) is non-negative to avoid complex values.

Let f(x) = √(2 - e^x). Give the interval(s) on which the function is continuous.

Learn how to find the interval where the function f(x) = √(2 - e^x) is continuous. This involves solving an inequality and understanding the domain restrictions for square root functions. Suitable for high school students in Grades 11-12.

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