To determine which statement is true about the function [tex]\( f(x) = \sqrt{x} \)[/tex], let's analyze its domain and range step by step.
### Step 1: Determine the Domain
The domain of a function is the set of all input values (x-values) for which the function is defined.
For the function [tex]\( f(x) = \sqrt{x} \)[/tex]:
- The square root function [tex]\( \sqrt{x} \)[/tex] is defined only when the value inside the square root is non-negative (i.e., [tex]\( x \geq 0 \)[/tex]).
- This is because the square root of a negative number is not a real number.
Thus, the domain of [tex]\( f(x) = \sqrt{x} \)[/tex] is all real numbers greater than or equal to 0.
### Step 2: Determine the Range
The range of a function is the set of all output values (y-values) that the function can produce.
For the function [tex]\( f(x) = \sqrt{x} \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = \sqrt{0} = 0 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x} \)[/tex] also increases.
- Since the square root function produces non-negative outputs for non-negative inputs, [tex]\( f(x) \)[/tex] will always be greater than or equal to 0.
Therefore, the range of [tex]\( f(x) = \sqrt{x} \)[/tex] is all real numbers greater than or equal to 0.
### Conclusion
- The domain of the function [tex]\( f(x) = \sqrt{x} \)[/tex] is all real numbers greater than or equal to 0.
- The range of the function [tex]\( f(x) = \sqrt{x} \)[/tex] is all real numbers greater than or equal to 0.
Given the provided options:
1. The domain of the graph is all real numbers. (False)
2. The range of the graph is all real numbers. (False)
3. The domain of the graph is all real numbers less than or equal to 0. (False)
4. The range of the graph is all real numbers greater than or equal to 0. (True)
Thus, the correct statement is:
The range of the graph is all real numbers greater than or equal to 0.
