Square Root Function

Definition of Square Root Function

A square root function is a type of function that involves square root operations. This function has the general form $f(x) = \sqrt{g(x)}$ where $g(x)$ is the function inside the square root sign.

The simplest form of a square root function is $f(x) = \sqrt{x}$ . This function takes an input value $x$ and produces the square root of that value.

Characteristics of Square Root Functions

Square root functions have several special characteristics that distinguish them from other functions:

Limited domain: Since the square root of negative numbers is not defined in real numbers, the domain of square root functions is limited to values that make the expression inside the square root non-negative.
Curved graph: The graph of a square root function is a curve that starts from a certain point and continues to rise at an increasingly slower rate.
Always non-negative values: The result of a square root function is always non-negative (≥ 0).

Domain and Range of Square Root Functions

To understand square root functions well, it's important to determine their domain and range.

Determining Domain

The domain of a square root function $f(x) = \sqrt{g(x)}$ is all values of $x$ that make $g(x) \geq 0$ .

Steps to determine domain:

Step	Explanation	Example: $f(x) = \sqrt{x-2}$
1	Identify the expression inside the square root	$g(x) = x-2$
2	Create the inequality $g(x) \geq 0$	$x-2 \geq 0$
3	Solve the inequality	$x \geq 2$
4	Write the domain	$D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}$

Determining Range

The range of a square root function is all possible output values that the function can produce.

For the function $f(x) = \sqrt{x-2}$ , since square roots always produce non-negative values, then:

R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}

Graph of Basic Square Root Function

Let's visualize the basic square root function $f(x) = \sqrt{x}$ .

Basic Square Root Function:

f(x) = \sqrt{x}

The graph of the basic square root function starts from point (0,0) and rises at an increasingly slower rate.

Transformations of Square Root Functions

Square root functions can undergo various transformations that change the shape and position of their graphs.

Horizontal Translation

The function $f(x) = \sqrt{x-h}$ shifts the graph of the basic square root function by $h$ units to the right (if $h > 0$ ) or to the left (if $h < 0$ ).

Horizontal Translation of Square Root Functions

Comparison of

f(x) = \sqrt{x}

g(x) = \sqrt{x-2}

, and

h(x) = \sqrt{x+2}

Vertical Translation

The function $f(x) = \sqrt{x} + k$ shifts the graph of the basic square root function by $k$ units upward (if $k > 0$ ) or downward (if $k < 0$ ).

Function	Transformation	Domain	Range
$f(x) = \sqrt{x}$	Basic function	$x \geq 0$	$y \geq 0$
$f(x) = \sqrt{x} + 2$	Shift 2 units upward	$x \geq 0$	$y \geq 2$
$f(x) = \sqrt{x} - 3$	Shift 3 units downward	$x \geq 0$	$y \geq -3$

Dilation

The function $f(x) = a\sqrt{x}$ with $a > 0$ causes vertical dilation on the graph of the square root function.

Vertical Dilation of Square Root Functions

Comparison of

f(x) = \sqrt{x}

g(x) = 2\sqrt{x}

, and

h(x) = \frac{1}{2}\sqrt{x}

General Form of Square Root Functions

The general form of a square root function that undergoes transformations is:

f(x) = a\sqrt{b(x-h)} + k

Where:

$a$ determines vertical dilation and reflection (if $a < 0$ )
$b$ determines horizontal dilation
$h$ determines horizontal translation
$k$ determines vertical translation

Steps to draw the graph:

Step	Action	Example: $f(x) = 2\sqrt{x-1} + 3$
1	Determine starting point	$x-1 = 0 \Rightarrow x = 1$ , $f(1) = 3$ , Point: (1, 3)
2	Determine domain	$x-1 \geq 0 \Rightarrow x \geq 1$
3	Create value table	Choose several values $x \geq 1$
4	Calculate function values	For $x = 2$ : $f(2) = 2\sqrt{1} + 3 = 5$
5	Plot points	Plot (1,3), (2,5), (5,7), etc.
6	Connect points	Create a smooth curve through the points

Square Root Function Equations

To solve equations involving square root functions, follow these steps:

Step	Explanation	Example: $\sqrt{2x+3} = 5$
1	Isolate the square root	Already isolated
2	Square both sides	$(\sqrt{2x+3})^2 = 5^2$
3	Simplify	$2x+3 = 25$
4	Solve	$2x = 22 \Rightarrow x = 11$
5	Verify	$\sqrt{2(11)+3} = \sqrt{25} = 5$ ✓

Square Root Function Inequalities

To solve square root function inequalities, pay attention to the domain and properties of square root functions.

Example: Solve $\sqrt{x-1} < 3$

\text{Condition: } x-1 \geq 0 \Rightarrow x \geq 1

\sqrt{x-1} < 3

x-1 < 9

x < 10

Combining with the domain condition: $1 \leq x < 10$

Practice Problems

Determine the domain and range of the function $f(x) = \sqrt{3x-6}$
Draw the graph of the function $g(x) = -\sqrt{x+4} + 2$
Solve the equation $\sqrt{x+5} + \sqrt{x} = 5$
A rocket is launched vertically. Its height after $t$ seconds is given by $h(t) = 100\sqrt{t}$ meters. What is the height of the rocket after 9 seconds?
Determine the value of $x$ that satisfies $2\sqrt{x-3} \geq 6$

Answer Key

Domain: $D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}$

Range: $R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}$

Drawing the graph $g(x) = -\sqrt{x+4} + 2$

Steps to draw:

Step	Explanation	Details for $g(x) = -\sqrt{x+4} + 2$
1	Identify transformations	$a = -1$ (reflection across x-axis), $h = -4$ (shift 4 units left), $k = 2$ (shift 2 units up)
2	Determine starting point	$x+4 = 0 \Rightarrow x = -4$ , $g(-4) = 0 + 2 = 2$ , Starting point: (-4, 2)
3	Determine domain	$x+4 \geq 0 \Rightarrow x \geq -4$
4	Determine range	Since $a = -1 < 0$ , the graph decreases from the starting point, so $y \leq 2$
5	Create value table	Choose values $x \geq -4$

Value table:

$x$	$x+4$	$\sqrt{x+4}$	$-\sqrt{x+4}$	$g(x) = -\sqrt{x+4} + 2$
-4	0	0	0	2
-3	1	1	-1	1
0	4	2	-2	0
5	9	3	-3	-1
12	16	4	-4	-2

Graph of

g(x) = -\sqrt{x+4} + 2

Graph of a square root function that undergoes reflection across the x-axis and translation.

$x = 4$ (Let $u = \sqrt{x}$ and $v = \sqrt{x+5}$ )
$h(9) = 100\sqrt{9} = 100 \times 3 = 300$ meters
$x \geq 12$ (from $\sqrt{x-3} \geq 3$ and condition $x \geq 3$ )

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