D6

Square Root Functions

Introduction to Square Root Functions

Square root functions are a fundamental type of function in algebra, typically written as f(x) = √x or in transformed forms such as f(x) = a√(x - h) + k. These functions are defined only for values of x that make the expression under the square root non-negative.

• Domain: The set of all x-values for which the function is defined (expression under the square root is non-negative).

• Range: The set of all possible output values (y-values) of the function.

• Point of Origin: The starting point of the graph, often where the function attains its minimum value.

• x-intercept: The value(s) of x where the function crosses the x-axis (f(x) = 0).

• y-intercept: The value of f(0), if defined.

Key Properties of Square Root Functions

General Form and Graph

The basic square root function is f(x) = √x. Its graph starts at the origin (0,0) and increases slowly as x increases.

• Domain: $x \geq 0$

• Range: $y \geq 0$

• Point of Origin: (0, 0)

• x-intercept: x = 0

• y-intercept: y = 0

Example: For $f(x) = \sqrt{x}$, the graph passes through (0,0), (1,1), (4,2), etc.

Inverse of Square Root Functions

The inverse of a square root function is a quadratic function. For $f(x) = \sqrt{x}$ (with $x \geq 0$), the inverse is $f^{-1}(x) = x^2$ (with $x \geq 0$).

• To find the inverse, swap x and y and solve for y:

$x = \sqrt{y}$ $y = x^2$

But restrict the domain to $x \geq 0$ to ensure the function is one-to-one.

Transformations of Square Root Functions

General Transformation

The general form is $f(x) = a\sqrt{x - h} + k$.

• Vertical shift: $k$ moves the graph up ($k > 0$) or down ($k < 0$).

• Horizontal shift: $h$ moves the graph right ($h > 0$) or left ($h < 0$).

• Vertical stretch/compression: $a$ stretches ($|a| > 1$) or compresses ($0 < |a| < 1$) the graph.

• Reflection: If $a < 0$, the graph is reflected over the x-axis.

Example: $f(x) = 2\sqrt{x - 3} + 1$ is shifted right by 3 units and up by 1 unit, and vertically stretched by a factor of 2.

Finding Domain, Range, and Intercepts

Domain and Range

• Set the expression under the square root $\geq 0$ to find the domain.

• Calculate the minimum value of $f(x)$ to find the range.

Example: For $f(x) = \sqrt{x - 2}$, domain is $x \geq 2$, range is $y \geq 0$.

Finding Intercepts

• x-intercept: Set $f(x) = 0$ and solve for x.

• y-intercept: Set $x = 0$ and solve for $f(0)$ (if in domain).

Example: For $f(x) = \sqrt{x - 2}$, x-intercept at $x = 2$, y-intercept is not defined (since $x = 0$ is not in the domain).

Worked Examples

Example 1: $f(x) = \sqrt{x - 3}$

• Domain: $x \geq 3$

• Range: $y \geq 0$

• Point of Origin: (3, 0)

• x-intercept: $x = 3$

• y-intercept: Not defined

Example 2: $f(x) = \sqrt{x + 2}$

• Domain: $x \geq -2$

• Range: $y \geq 0$

• Point of Origin: (-2, 0)

• x-intercept: $x = -2$

• y-intercept: $f(0) = \sqrt{2}$

Example 3: $f(x) = \sqrt{-(x - 1)}$

• Domain: $x \leq 1$

• Range: $y \geq 0$

• Point of Origin: (1, 0)

• x-intercept: $x = 1$

• y-intercept: $f(0) = \sqrt{1}$

Table: Properties of Square Root Functions

Function	Point of Origin	Domain	Range	x-intercept	y-intercept	Sketch of Graph
$f(x) = \sqrt{x - 3} + 1$	(3, 1)	$[3, \infty)$	$[1, \infty)$	3	None	Starts at (3,1), rises rightward
$f(x) = \sqrt{x + 2} - 3$	(-2, -3)	$[-2, \infty)$	$[-3, \infty)$	-2	$-3 + \sqrt{2}$	Starts at (-2,-3), rises rightward
$f(x) = \sqrt{-(x - 1)}$	(1, 0)	$(-\infty, 1]$	$[0, \infty)$	1	1	Starts at (1,0), rises leftward

Solving Equations Involving Square Roots

Steps for Solving

1. Isolate the square root on one side of the equation.

2. Square both sides to eliminate the square root.

3. Solve the resulting equation for x.

4. Check all solutions in the original equation to avoid extraneous solutions.

Example: Solve $\sqrt{x - 3} = 2$

• Isolate: $\sqrt{x - 3} = 2$

• Square both sides: $x - 3 = 4$

• Solve: $x = 7$

• Check: $\sqrt{7 - 3} = \sqrt{4} = 2$ (valid)

Common Errors and Extraneous Solutions

• Squaring both sides can introduce extraneous solutions. Always check proposed solutions in the original equation.

• The square root of a negative number is not a real number; restrict solutions to the domain of the function.

Example: $\sqrt{x - 3} = -2$ has no real solution, since the square root function cannot yield a negative result.

Practice Problems

1. Complete the table for each square root function (see above table for format).

2. Given the graph of a square root function, write its equation by identifying shifts and transformations.

3. Find the equation of a square root function given its domain, range, and a point of origin.

4. Solve equations involving square roots and check all proposed solutions for validity.

Summary

• Square root functions are defined for non-negative radicands and have a characteristic graph shape.

• Transformations shift, stretch, or reflect the graph.

• Always check for extraneous solutions when solving equations involving square roots.