To determine the domain and range of the function [tex]\( f(x) = \sqrt{x - 4} \)[/tex], let's analyze the function step-by-step.
### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the square root function [tex]\( \sqrt{x - 4} \)[/tex], the expression inside the square root must be non-negative because the square root of a negative number is not a real number.
To find the condition for the domain:
[tex]\[ x - 4 \ge 0 \][/tex]
Solving this inequality:
[tex]\[ x \ge 4 \][/tex]
Thus, the domain of the function is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 4. In interval notation, this can be written as:
[tex]\[ [4, \infty) \][/tex]
### Step 2: Determine the Range
The range of a function is the set of all possible output values (y-values). For the function [tex]\( f(x) = \sqrt{x - 4} \)[/tex], let's analyze its behavior.
Since [tex]\( f(x) \)[/tex] represents the principal square root, it will always yield non-negative values. The smallest value inside the square root is achieved when [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = \sqrt{4 - 4} = \sqrt{0} = 0 \][/tex]
As [tex]\( x \)[/tex] increases beyond 4, the value of [tex]\( \sqrt{x - 4} \)[/tex] also increases. There's no upper bound to the values [tex]\( f(x) \)[/tex] can take because as [tex]\( x \)[/tex] grows larger, [tex]\( x - 4 \)[/tex] also grows larger, and thus [tex]\( \sqrt{x - 4} \)[/tex] can become arbitrarily large.
Therefore, the range of the function is all real numbers [tex]\( y \)[/tex] such that [tex]\( y \)[/tex] is greater than or equal to 0. In interval notation, this can be represented as:
[tex]\[ [0, \infty) \][/tex]
### Final Answer
- Domain: [tex]\([4, \infty)\)[/tex]
- Range: [tex]\([0, \infty)\)[/tex]
So, the domain of the function [tex]\( f(x) = \sqrt{x - 4} \)[/tex] is [tex]\([4, \infty)\)[/tex] and the range is [tex]\([0, \infty)\)[/tex].
