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.