The Function $y = \sqrt{x}$ and Its Graph
Big Idea: The square root function is one of the most important functions in mathematics. It "undoes" squaring, and its graph has a beautiful curved shape that reveals deep properties about how numbers relate to their square roots.
📐 Formal Definition
The square root function is defined as:
$$y = \sqrt{x}$$
This means: "y is the non-negative number whose square equals x."
In other words, if $y = \sqrt{x}$, then $y^2 = x$ and $y \geq 0$.
Domain and Range
- Domain: $x \geq 0$ (all non-negative real numbers)
We cannot take the square root of a negative number in the real number system. - Range: $y \geq 0$ (all non-negative real numbers)
The square root function always outputs non-negative values.
📊 Table of Key Values
| $x$ | $y = \sqrt{x}$ | Exact Value |
|---|---|---|
| 0 | $\sqrt{0} = 0$ | 0 |
| 1 | $\sqrt{1} = 1$ | 1 |
| 2 | $\sqrt{2}$ | ≈ 1.41 |
| 4 | $\sqrt{4} = 2$ | 2 |
| 9 | $\sqrt{9} = 3$ | 3 |
| 16 | $\sqrt{16} = 4$ | 4 |
| 25 | $\sqrt{25} = 5$ | 5 |