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Functions and Their Modeling

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)=g(x)f(x) = \sqrt{g(x)}f(x)=g(x)​ where g(x)g(x)g(x) is the function inside the square root sign.

The simplest form of a square root function is f(x)=xf(x) = \sqrt{x}f(x)=x​. This function takes an input value xxx 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:

  1. 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.

  2. 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.

  3. 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)=g(x)f(x) = \sqrt{g(x)}f(x)=g(x)​ is all values of xxx that make g(x)≥0g(x) \geq 0g(x)≥0.

Steps to determine domain:

StepExplanationExample: f(x)=x−2f(x) = \sqrt{x-2}f(x)=x−2​
1Identify the expression inside the square rootg(x)=x−2g(x) = x-2g(x)=x−2
2Create the inequality g(x)≥0g(x) \geq 0g(x)≥0x−2≥0x-2 \geq 0x−2≥0
3Solve the inequalityx≥2x \geq 2x≥2
4Write the domainDf={x∣x≥2,x∈R}D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}Df​={x∣x≥2,x∈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)=x−2f(x) = \sqrt{x-2}f(x)=x−2​, since square roots always produce non-negative values, then:

Rf={y∣y≥0,y∈R}R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}Rf​={y∣y≥0,y∈R}

Graph of Basic Square Root Function

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

Basic Square Root Function: f(x)=xf(x) = \sqrt{x}f(x)=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)=x−hf(x) = \sqrt{x-h}f(x)=x−h​ shifts the graph of the basic square root function by hhh units to the right (if h>0h > 0h>0) or to the left (if h<0h < 0h<0).

Horizontal Translation of Square Root Functions
Comparison of f(x)=xf(x) = \sqrt{x}f(x)=x​, g(x)=x−2g(x) = \sqrt{x-2}g(x)=x−2​, and h(x)=x+2h(x) = \sqrt{x+2}h(x)=x+2​.

Vertical Translation

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

FunctionTransformationDomainRange
f(x)=xf(x) = \sqrt{x}f(x)=x​Basic functionx≥0x \geq 0x≥0y≥0y \geq 0y≥0
f(x)=x+2f(x) = \sqrt{x} + 2f(x)=x​+2Shift 2 units upwardx≥0x \geq 0x≥0y≥2y \geq 2y≥2
f(x)=x−3f(x) = \sqrt{x} - 3f(x)=x​−3Shift 3 units downwardx≥0x \geq 0x≥0y≥−3y \geq -3y≥−3

Dilation

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

Vertical Dilation of Square Root Functions
Comparison of f(x)=xf(x) = \sqrt{x}f(x)=x​, g(x)=2xg(x) = 2\sqrt{x}g(x)=2x​, and h(x)=12xh(x) = \frac{1}{2}\sqrt{x}h(x)=21​x​.

General Form of Square Root Functions

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

f(x)=ab(x−h)+kf(x) = a\sqrt{b(x-h)} + kf(x)=ab(x−h)​+k

Where:

  • aaa determines vertical dilation and reflection (if a<0a < 0a<0 )
  • bbb determines horizontal dilation
  • hhh determines horizontal translation
  • kkk determines vertical translation

Steps to draw the graph:

StepActionExample: f(x)=2x−1+3f(x) = 2\sqrt{x-1} + 3f(x)=2x−1​+3
1Determine starting pointx−1=0⇒x=1x-1 = 0 \Rightarrow x = 1x−1=0⇒x=1, f(1)=3f(1) = 3f(1)=3, Point: (1, 3)
2Determine domainx−1≥0⇒x≥1x-1 \geq 0 \Rightarrow x \geq 1x−1≥0⇒x≥1
3Create value tableChoose several values x≥1x \geq 1x≥1
4Calculate function valuesFor x=2x = 2x=2: f(2)=21+3=5f(2) = 2\sqrt{1} + 3 = 5f(2)=21​+3=5
5Plot pointsPlot (1,3), (2,5), (5,7), etc.
6Connect pointsCreate a smooth curve through the points

Square Root Function Equations

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

StepExplanationExample: 2x+3=5\sqrt{2x+3} = 52x+3​=5
1Isolate the square rootAlready isolated
2Square both sides(2x+3)2=52(\sqrt{2x+3})^2 = 5^2(2x+3​)2=52
3Simplify2x+3=252x+3 = 252x+3=25
4Solve2x=22⇒x=112x = 22 \Rightarrow x = 112x=22⇒x=11
5Verify2(11)+3=25=5\sqrt{2(11)+3} = \sqrt{25} = 52(11)+3​=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 x−1<3\sqrt{x-1} < 3x−1​<3

Condition: x−1≥0⇒x≥1\text{Condition: } x-1 \geq 0 \Rightarrow x \geq 1Condition: x−1≥0⇒x≥1
x−1<3\sqrt{x-1} < 3x−1​<3
x−1<9x-1 < 9x−1<9
x<10x < 10x<10

Combining with the domain condition: 1≤x<101 \leq x < 101≤x<10

Practice Problems

  1. Determine the domain and range of the function f(x)=3x−6f(x) = \sqrt{3x-6}f(x)=3x−6​

  2. Draw the graph of the function g(x)=−x+4+2g(x) = -\sqrt{x+4} + 2g(x)=−x+4​+2

  3. Solve the equation x+5+x=5\sqrt{x+5} + \sqrt{x} = 5x+5​+x​=5

  4. A rocket is launched vertically. Its height after ttt seconds is given by h(t)=100th(t) = 100\sqrt{t}h(t)=100t​ meters. What is the height of the rocket after 9 seconds?

  5. Determine the value of xxx that satisfies 2x−3≥62\sqrt{x-3} \geq 62x−3​≥6

Answer Key

  1. Domain: Df={x∣x≥2,x∈R}D_f = \{x \mid x \geq 2, x \in \mathbb{R}\}Df​={x∣x≥2,x∈R}

    Range: Rf={y∣y≥0,y∈R}R_f = \{y \mid y \geq 0, y \in \mathbb{R}\}Rf​={y∣y≥0,y∈R}

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

    Steps to draw:

    StepExplanationDetails for g(x)=−x+4+2g(x) = -\sqrt{x+4} + 2g(x)=−x+4​+2
    1Identify transformationsa=−1a = -1a=−1 (reflection across x-axis), h=−4h = -4h=−4 (shift 4 units left), k=2k = 2k=2 (shift 2 units up)
    2Determine starting pointx+4=0⇒x=−4x+4 = 0 \Rightarrow x = -4x+4=0⇒x=−4, g(−4)=0+2=2g(-4) = 0 + 2 = 2g(−4)=0+2=2, Starting point: (-4, 2)
    3Determine domainx+4≥0⇒x≥−4x+4 \geq 0 \Rightarrow x \geq -4x+4≥0⇒x≥−4
    4Determine rangeSince a=−1<0a = -1 < 0a=−1<0, the graph decreases from the starting point, so y≤2y \leq 2y≤2
    5Create value tableChoose values x≥−4x \geq -4x≥−4

    Value table:

    xxxx+4x+4x+4x+4\sqrt{x+4}x+4​−x+4-\sqrt{x+4}−x+4​g(x)=−x+4+2g(x) = -\sqrt{x+4} + 2g(x)=−x+4​+2
    -40002
    -311-11
    042-20
    593-3-1
    12164-4-2
    Graph of g(x)=−x+4+2g(x) = -\sqrt{x+4} + 2g(x)=−x+4​+2
    Graph of a square root function that undergoes reflection across the x-axis and translation.

  3. x=4x = 4x=4 (Let u=xu = \sqrt{x}u=x​ and v=x+5v = \sqrt{x+5}v=x+5​ )

  4. h(9)=1009=100×3=300h(9) = 100\sqrt{9} = 100 \times 3 = 300h(9)=1009​=100×3=300 meters

  5. x≥12x \geq 12x≥12 (from x−3≥3\sqrt{x-3} \geq 3x−3​≥3 and condition x≥3x \geq 3x≥3)

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  • Square Root FunctionExplore square root functions with interactive graphs, domain/range analysis, transformations, and equations. Master graphing techniques through rocket height problems.
On this page
  • Definition of Square Root Function
    • Characteristics of Square Root Functions
  • Domain and Range of Square Root Functions
    • Determining Domain
    • Determining Range
  • Graph of Basic Square Root Function
  • Transformations of Square Root Functions
    • Horizontal Translation
    • Vertical Translation
    • Dilation
  • General Form of Square Root Functions
  • Square Root Function Equations
  • Square Root Function Inequalities
  • Practice Problems
    • Answer Key
  • Comments
  • Report
  • Source code