By Luis Olortegui — Updated Aug 30, 2022
In mathematics, a function is expressed as y = f(x), where x is the independent variable (input) and y is the dependent variable (output). The set of all possible input values is called the domain, while the set of all possible output values is called the range.
For a square‑root function, the output is defined by the equation y² = x. Because a square root cannot be taken of a negative number, the expression inside the root must be non‑negative, which imposes restrictions on both the domain and the range.
Begin by stating the full equation of the square‑root function. For example:
f(x) = y = √(x³ – 8)
Set the expression inside the root greater than or equal to zero and solve for x:
x³ – 8 ≥ 0 ⇒ x³ ≥ 8 ⇒ x ≥ 2
Thus, the domain is [2, ∞). All input values less than 2 would make the expression inside the root negative and are therefore excluded.
With the domain established, evaluate the function at key points to observe how the output behaves. Starting at the left boundary of the domain:
As x increases, the square‑root output increases without bound. Therefore, the range is [0, ∞).
In summary, the square‑root function f(x) = √(x³ – 8) has a domain of all real numbers greater than or equal to 2 and a range of all real numbers greater than or equal to 0.
