Certainly! Let's analyze the function [tex]\( g(x) = \sqrt{x} \)[/tex].
### Finding the Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept without resulting in an undefined expression. For the function [tex]\( g(x) = \sqrt{x} \)[/tex], the expression inside the square root must be non-negative because the square root of a negative number is not defined in the set of real numbers.
So, we need to find the values of [tex]\( x \)[/tex] for which the expression inside the square root is non-negative:
[tex]\[ x \geq 0 \][/tex]
In interval notation, the domain is represented as:
[tex]\[ [0, \infty) \][/tex]
### Finding the Range:
The range of a function is the set of all possible output values (y-values) that the function can produce. For the function [tex]\( g(x) = \sqrt{x} \)[/tex], we need to determine the possible values for [tex]\( y = \sqrt{x} \)[/tex].
When [tex]\( x = 0 \)[/tex], we get:
[tex]\[ g(0) = \sqrt{0} = 0 \][/tex]
As [tex]\( x \)[/tex] increases from 0 to positive infinity ([tex]\( \infty \)[/tex]), [tex]\( \sqrt{x} \)[/tex] increases as well and can theoretically take on any non-negative real value.
Thus, the range of [tex]\( g(x) = \sqrt{x} \)[/tex] is the set of all non-negative real numbers:
[tex]\[ y \geq 0 \][/tex]
In interval notation, the range is represented as:
[tex]\[ [0, \infty) \][/tex]
### Final Answer:
So, the domain and range of the function [tex]\( g(x) = \sqrt{x} \)[/tex] are both:
[tex]\[ [0, \infty) \][/tex]
Expressed together in interval notation:
Domain: [tex]\([0, \infty)\)[/tex]
Range: [tex]\([0, \infty)\)[/tex]
