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The function given is:

\[
f(x) = \left|\sqrt{x^2 - 1}\right| + 4
\]

To find the range of this function, let's analyze it step by step.

### Step 1: Analyze the expression inside the absolute value
The expression inside the square root is \( x^2 - 1 \). Since the square root function is defined only for non-negative values, we have:

\[
x^2 - 1 \geq 0
\]

This inequality holds when:

\[
x^2 \geq 1 \quad \Rightarrow \quad |x| \geq 1 \quad \Rightarrow \quad x \leq -1 \text{ or } x \geq 1
\]

So, the function \( f(x) \) is defined for \( x \leq -1 \) and \( x \geq 1 \).

### Step 2: Simplify the function
Since \( x^2 - 1 \geq 0 \), the square root is always non-negative. Thus, the absolute value \( \left|\sqrt{x^2 - 1}\right| \) simply equals \( \sqrt{x^2 - 1} \). Therefore:

\[
f(x) = \sqrt{x^2 - 1} + 4
\]

### Step 3: Determine the minimum and maximum values of \( f(x) \)
The minimum value of \( \sqrt{x^2 - 1} \) occurs when \( x^2 \) is at its minimum within the domain \( |x| \geq 1 \). For \( |x| = 1 \), we have:

\[
\sqrt{1^2 - 1} = \sqrt{0} = 0
\]

So, the minimum value of \( f(x) \) is:

\[
f(x)_{\text{min}} = 0 + 4 = 4
\]

As \( |x| \) increases (i.e., \( |x| > 1 \)), \( x^2 \) increases, and thus \( \sqrt{x^2 - 1} \) increases as well. Since the square root function tends to infinity as \( x^2 \) becomes large, \( f(x) \) has no upper bound.

### Conclusion
The range of \( f(x) \) is:

\[
\text{Range of } f(x) = [4, \infty)
\]

Would you like more details or have any questions? Here are some related questions you might consider:

1. How would the range of \( f(x) \) change if we subtracted 4 instead of adding it?
2. What is the domain of the function \( f(x) \)?
3. How would the range be affected if the function had \( \sqrt{1 - x^2} \) instead?
4. What are the properties of square root functions and their impact on range?
5. How would the range be different if the absolute value were applied to the entire expression?

**Tip:** When finding the range of a function, consider the behavior of the function at the boundaries of its domain as well as the function's tendency (e.g., does it go to infinity or approach a certain value?).

Explore how to find the range of the function f(x) = |√(x^2 - 1)| + 4 step-by-step. Learn about absolute value, square root functions, and function range analysis. Suitable for high school students studying advanced algebra.

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