Let's carefully analyze the function [tex]\( f(x) = \sqrt{x} \)[/tex] and determine which statement about it is true.
1. Domain of [tex]\( f(x) = \sqrt{x} \)[/tex]:
- The domain of a function refers to all possible input values (x-values) that the function can accept.
- The square root function, [tex]\( \sqrt{x} \)[/tex], is only defined for [tex]\( x \)[/tex] values that are greater than or equal to 0 because the square root of a negative number is not a real number.
- Therefore, the domain of [tex]\( f(x) = \sqrt{x} \)[/tex] is all real numbers [tex]\( x \geq 0 \)[/tex].
2. Range of [tex]\( f(x) = \sqrt{x} \)[/tex]:
- The range of a function refers to all possible output values (y-values) that the function can produce.
- For the square root function [tex]\( f(x) = \sqrt{x} \)[/tex], when [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]. As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] produces positive values (square roots of those numbers).
- Therefore, the range of [tex]\( f(x) = \sqrt{x} \)[/tex] is all real numbers [tex]\( y \geq 0 \)[/tex].
Now, let's examine the given statements one by one:
1. The domain of the graph is all real numbers.
- This statement is false because the domain is all real numbers [tex]\( x \geq 0 \)[/tex], not all real numbers.
2. The range of the graph is all real numbers.
- This statement is false because the range is all real numbers [tex]\( y \geq 0 \)[/tex], not all real numbers.
3. The domain of the graph is all real numbers less than or equal to 0.
- This statement is false because the domain is all real numbers [tex]\( x \geq 0 \)[/tex], not [tex]\( x \leq 0 \)[/tex].
4. The range of the graph is all real numbers greater than or equal to 0.
- This statement is true because, as analyzed, the range of [tex]\( f(x) = \sqrt{x} \)[/tex] is [tex]\( y \geq 0 \)[/tex].
Thus, the correct statement is:
The range of the graph is all real numbers greater than or equal to 0.
