Square Root Function

Algebra-1

1. Fundamental Concepts

Definition: A square root function is a function of the form $$f(x) = \sqrt{x}$$ where the output is the non-negative value which, when multiplied by itself, gives the input.
Domain: The domain of the square root function is all real numbers greater than or equal to zero ( $$x \geq 0$$ ).
Range: The range of the square root function is all non-negative real numbers ( $$f(x) \geq 0$$ ).

2. Key Concepts

Evaluating Square Roots: $$\sqrt{a^2} = |a|$$

Solving Equations: To solve $$\sqrt{x} = k$$ , square both sides: $$x = k^2$$

Graphing: The graph of $$y = \sqrt{x}$$ starts at (0, 0) and increases as x increases.

3. Examples

Example 1 (Basic)

Problem: Evaluate $$\sqrt{16}$$

Step-by-Step Solution:

The square root of 16 is the number that, when squared, equals 16. This number is 4.

Validation: Substitute into original expression → $$\sqrt{16} = 4$$ ✓

Example 2 (Intermediate)

Problem: Solve for x in $$\sqrt{x} = 5$$

Step-by-Step Solution:

Square both sides of the equation: $$(\sqrt{x})^2 = 5^2$$
This simplifies to: $$x = 25$$

Validation: Substitute x=25 into original equation → $$\sqrt{25} = 5$$ ✓

4. Problem-Solving Techniques

Isolate the Square Root: Always isolate the square root term on one side of the equation before squaring both sides.
Check Solutions: After solving, substitute the solutions back into the original equation to ensure they are valid (i.e., they do not make the radicand negative).
Graphical Interpretation: Use graphs to visualize the behavior of square root functions and understand their domains and ranges.