Square Root Function Properties
This Grade 11 math skill from enVision Algebra 1 defines the properties of the square root function. The square root function f(x) = √x has a domain of [0, ∞) — all non-negative real numbers — because square roots of negative numbers are not real. Its range is also [0, ∞). The graph starts at the origin (0, 0) and increases, but at a decreasing rate, curving to the right. Students learn to identify domain and range, evaluate the function, and distinguish it from other parent functions like linear, quadratic, and absolute value functions.
Key Concepts
Property | Function | Definition | Domain | Range | | | | | | | Square Root Function | $f(x) = \sqrt{x}$ | $[0, \infty)$ | $[0, \infty)$ |.
Examples The function $f(x) = \sqrt{x}$ is undefined for negative inputs, so its domain starts at 0. For example, $f(25) = 5$ and $f(9) = 3$. Since we can only take the square root of non negative numbers, the domain is $[0, \infty)$. For instance, $f(0) = 0$, $f(4) = 2$, and $f(16) = 4$. The range is $[0, \infty)$ because the square root function can never produce a negative output value. The smallest possible output is $f(0) = 0$.
Explanation The square root function $f(x) = \sqrt{x}$ starts at the point $(0,0)$ and accepts only non negative inputs. This restriction exists because we cannot take the square root of negative numbers in the real number system. The function produces only non negative outputs, creating a curved graph that increases gradually as $x$ increases.
Common Questions
What is the square root function?
The square root function f(x) = √x returns the principal (non-negative) square root of x. It is defined only for non-negative inputs because square roots of negative numbers are not real numbers.
What is the domain of the square root function?
The domain of f(x) = √x is [0, ∞) — all non-negative real numbers. You cannot take the square root of a negative number and get a real result, so negative inputs are excluded.
What is the range of the square root function?
The range of f(x) = √x is [0, ∞) — all non-negative real numbers. Since √x ≥ 0 for all valid inputs, the function only outputs non-negative values.
What does the graph of the square root function look like?
The graph of f(x) = √x starts at the origin (0, 0), curves upward and to the right, and increases at a decreasing rate — getting flatter as x gets larger. It only exists in the first quadrant.
How does the square root function compare to the quadratic function?
The square root function f(x) = √x and the quadratic function f(x) = x² are inverse functions of each other (on the non-negative domain). The square root undoes squaring, and squaring undoes the square root.