D6

Square Root Functions

Introduction to Square Root Functions

Square root functions are a type of radical function commonly studied in College Algebra. They have unique properties regarding their domain, range, intercepts, and transformations. Understanding these characteristics is essential for graphing and analyzing such functions.

• Definition: A square root function is typically written as $f(x) = \sqrt{x}$ or in the more general form $f(x) = a\sqrt{x-h} + k$.

• Key Properties:

  • The graph starts at a specific point called the point of origin.

  • The function is only defined for values of $x$ where the expression under the square root is non-negative.

  • The graph increases slowly and is always above or on the $x$-axis.

Domain and Range of Square Root Functions

The domain and range of a square root function depend on the expression inside the radical and any transformations applied.

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

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

• Example: For $f(x) = \sqrt{x-2}$, the domain is $[2, \infty)$ and the range is $[0, \infty)$.

Finding Intercepts and Point of Origin

Intercepts and the point of origin are important for graphing and understanding the behavior of square root functions.

• Point of Origin: The starting point of the graph, found by setting the expression under the square root to zero.

• x-intercept: The value of $x$ where $f(x) = 0$.

• y-intercept: The value of $f(0)$, if $0$ is in the domain.

• Example: For $f(x) = \sqrt{x-3}$, the point of origin is $(3, 0)$, and the $x$-intercept is $x=3$.

Inverse of Square Root Functions

Square root functions are the inverse of quadratic functions restricted to non-negative $x$ values.

• Inverse Function: If $f(x) = x^2$ for $x \geq 0$, then its inverse is $f^{-1}(x) = \sqrt{x}$ for $x \geq 0$.

• Graphical Relationship: The graph of a function and its inverse are reflections across the line $y = x$.

• Example: The inverse of $f(x) = x^2$ (for $x \geq 0$) is $f^{-1}(x) = \sqrt{x}$.

Transformations of Square Root Functions

Square root functions can be shifted and stretched using transformations.

• Vertical Shift: $f(x) = \sqrt{x} + k$ shifts the graph up by $k$ units.

• Horizontal Shift: $f(x) = \sqrt{x-h}$ shifts the graph right by $h$ units.

• Reflection: $f(x) = -\sqrt{x}$ reflects the graph across the $x$-axis.

• Example: $f(x) = \sqrt{x-2} + 3$ is shifted right by 2 units and up by 3 units.

Solving Equations Involving Square Root Functions

To solve equations with square root functions, isolate the radical and square both sides to eliminate the square root.

• Step 1: Isolate the square root on one side of the equation.

• Step 2: Square both sides to remove the radical.

• Step 3: Solve the resulting equation and check for extraneous solutions.

• Example: Solve $\sqrt{x-3} = 2$.

  • Square both sides: $x-3 = 4$

  • Solve: $x = 7$

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

Common Mistakes and Extraneous Solutions

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

• Extraneous Solution: A solution that does not satisfy the original equation after squaring both sides.

• Example: Solving $\sqrt{x-3} = -2$ yields $x-3 = 4$, $x = 7$, but $\sqrt{7-3} = 2$, not $-2$; so there is no solution.

Tables: Properties of Square Root Functions

The following table summarizes the point of origin, domain, range, and intercepts for several square root functions:

Function	Point of Origin	Domain	Range	x-intercept	y-intercept	Sketch of Graph
$f(x) = \sqrt{x-5} + 1$	(5, 1)	$[5, \infty)$	$[1, \infty)$	None	None	Starts at (5,1), rises slowly to the right
$f(x) = \sqrt{x+3} - 2$	(-3, -2)	$[-3, \infty)$	$[-2, \infty)$	$x = -3$	None	Starts at (-3,-2), rises slowly to the right
$f(x) = \sqrt{x-1} - 3$	(1, -3)	$[1, \infty)$	$[-3, \infty)$	$x = 1$	None	Starts at (1,-3), rises slowly to the right

Practice Problems

• Find the equation for a square root function given its graph and properties.

• Solve equations involving square root functions and check for extraneous solutions.

• Sketch the graph of $f(x) = \sqrt{x-5} + 1$ and identify its domain, range, and intercepts.

Summary

• Square root functions are defined for non-negative radicands and have a distinct starting point.

• Transformations shift and reflect the graph.

• Solving equations requires isolating the radical and checking for extraneous solutions.

• Understanding domain, range, and intercepts is essential for graphing and analyzing these functions.