$$  $$$$  $$$$ f\left(x\right)=\sqrt{x-4} $$

Let's analyze the function $ f(x) = \sqrt{x - 4} $ to determine its domain and range.

### **Domain**:
The function $ f(x) = \sqrt{x - 4} $ involves a square root. The expression inside the square root, $ x - 4 $, must be non-negative because the square root of a negative number is not defined in the set of real numbers.

Therefore, we require:
\[
x - 4 \geq 0
\]

Solving this inequality:
\[
x \geq 4
\]

So, the domain of the function is:
\[
\text{Domain}: \ [4, \infty)
\]

### **Range**:
The square root function $ \sqrt{y} $ always yields non-negative results, i.e., $ \sqrt{y} \geq 0 $ for any $ y \geq 0 $.

Since $ x - 4 \geq 0 $, the output of $ f(x) = \sqrt{x - 4} $ is also non-negative. The smallest value of $ f(x) $ occurs when $ x = 4 $, which gives $ f(4) = \sqrt{4 - 4} = 0 $. As $ x $ increases, $ f(x) $ increases without bound.

Thus, the range of the function is:
\[
\text{Range}: \ [0, \infty)
\]

### **Summary**:
For the function $ f(x) = \sqrt{x - 4} $:
- **Domain**: $ [4, \infty) $
- **Range**: $ [0, \infty) $

Would you like any further explanation or have more questions?

Here are 5 related questions:
1. Find the domain and range of $ f(x) = \sqrt{9 - x^2} $.
2. What is the domain of $ f(x) = \frac{1}{\sqrt{x + 3}} $?
3. Determine the domain of $ f(x) = \sqrt{2x + 5} $.
4. Find the range of $ f(x) = \sqrt{x} + 2 $.
5. What are the domain and range of $ f(x) = \sqrt{(x-1)(x-5)} $?

**Tip:** When finding the domain of a square root function, always ensure the expression under the square root is non-negative.

Explore the domain and range of the function f(x) = sqrt(x - 4) with a comprehensive explanation. Learn about square root functions and how to determine the permissible input values and output range. Suitable for advanced high school students.

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