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Which statement is true about the function [tex]f(x) = \sqrt{x}[/tex]?

A. The domain of the graph is all real numbers.
B. The range of the graph is all real numbers.
C. The domain of the graph is all real numbers less than or equal to 0.
D. The range of the graph is all real numbers greater than or equal to 0.

Sagot :

GinnyAnswer
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.
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