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Square Root Function
The square root function involves the square root symbol √ (which is read as "square root of"). The square root of a number 'x' is a number 'y' such that y^{2} = x. i.e., if y^{2} = x ⇒ y = √x. i.e., if 'x' is the square of 'y' then 'y' is the square root of 'x'. Some examples are:
 2^{2} = 4 ⇒ √4 = 2
 4^{2} = 16 ⇒ √16 = 4
We know that the square root of a number can be either positive or negative. i.e.,√4 = ±2. But while defining the square root function, we restrict its range to be the set of all positive real numbers (otherwise it won't become a function at all), and hence in the case of the square root function, the result is always positive. Let us use all these facts to understand the square root function.
What is Square Root Function?
The square root function is basically of the form f(x) = √x. i.e., the parent square root function is f(x) = √x. This is the inverse of the square function g(x) = x^{2} as the square and square root are the inverse operations of each other. As the square root function is increasing (as the values of f(x) increase with the increase of values of x) and as it is oneone, it is a bijection and so it has an inverse. The graphs square root function f(x) = √x and its inverse g(x) = x^{2} over the domain [0, ∞) and the range [0, ∞) are symmetric with respect to the line y = x as shown in the figure below.
f(x) = √x is the parent square root function but when the transformations are applied to it, it may look like f(x) = a√(b(x  h)) + k, where a, b, h, and k are numbers such that
 'a' is the vertical dilation
 'b' is the horizontal dilation
 'h' is the horizontal translation
 'k' is the vertical translation
Domain and Range of Square Root Function
The square root of a negative number is NOT a real number. i.e., the square root function cannot accept negative numbers as inputs. i.e.,
 The domain of the square root function f(x) = √x is the set of all nonnegative real numbers. i.e., the square root function domain is [0, ∞). Note that it includes 0 as well in the domain.
In general, the square root of a number can be either positive or negative. i.e., √25 = 5 or 5 as 5^{2} = 25 and (5)^{2} = 25. But the range of the square root function (i.e., its yvalues) is restricted to only positive numbers, because otherwise, it fails the vertical line test and it won't be a function if an input has two outputs. Thus,
 The range of the square root function f(x) = √x is also the same as its domain [0, ∞). Note that it includes 0 as well in the range.
Square Root Graph
We have already seen how a square root graph looks like. But now we will see how to graph the square root of x. We have already seen that the domain and the range of the parent square root function f(x) = √x is the set of all nonnegative real numbers. Thus, the square root graph of f(x) = √x lies only in the first quadrant. We can draw its graph by constructing a table of values with some random values of x (from the domain [0, ∞), and then computing the corresponding values of y by substituting each x into y = √x. Then we can get some points that we will plot on the coordinate plane and join all of them by a curve.
x  y 

0  √0 = 0 
1  √1 = 1 
4  √4 = 2 
Note that when some transformations are applied to the graph, the graph may not lie in the first quadrant itself.
Graphing Any Square Root Function
We have seen how to graph the parent square root function f(x) = √x. Here are the steps that are useful in graphing any square root function that is of the form f(x) = a√(b(x  h)) + k in general.
 Step 1: Identify the domain of the function by setting "the expression inside the square root" to greater than or equal to 0 and solving for x.
 Step 2: The range of any square root function is always y ≥ k where 'k' is the vertical translation of the function f(x) = a√(b(x  h)) + k.
 Step 3: Construct a table of values with two columns x and y, take some random numbers for x (from the domain only) starting from the first value of the domain, substitute them in the given function and find the corresponding values of y.
 Step 4: Plot all the points on the plane and connect them by a curve and also extend the curve following the same trend.
Note: Computing the xintercept and yintercept would also help in graphing the square root function.
Example: Graph the square root function f(x) = √(x  2) + 3.
Solution:
To find its domain, x  2 ≥ 0 ⇒ x ≥ 2.
Its vertical shift is 3 and hence its range is y ≥ 3.
Now, we will construct a table with some values greater than 2 (as the domain is x ≥ 2). Choose some values for x such that √(x  2) is a perfect square so that the calculation becomes easier.
x  y 

2  √(2  2) + 3 = 0 + 3 = 3 
3  √(3  2) + 3 = 1 + 3 = 4 
6  √(6  2) + 3 = 2 + 3 = 5 
11  √(11  2) + 3 = 3 + 3 = 6 
Now, plot these points and join them by a curve.
We can also graph the square root function by applying the transformations on the parent square root graph f(x) = √x.
Properties of Square Root Function
Here are the important points/properties that are to be noted about the square root function f(x) = √x.
 Its domain is [0, ∞).
 Its range is [0, ∞).
 It has no relative maxima but it has a minimum at (0, 0).
 A square root function has no asymptotes.
 It is an increasing function throughout its domain [0, ∞).
 The square root function f(x) = √x has critical point at (0, 0) and it has no inflection points.
Square Root Function Examples

Example 1: Find the outputs if the square root function f(x) = 2√(x + 1) + 7 for the inputs a) x = 0 b) x = 3 c) x = 8?
Solution:
We just substitute each input in the function to get the corresponding output.
a) f(0) = 2√(0 + 1) + 7 = 2(1) + 7 = 9
b) f(3) = 2√(3 + 1) + 7 = 2(2) + 7 = 11
c) f(8) = 2√(8 + 1) + 7 = 2(3) + 7 = 13
Answer: a) f(0) = 9 b) f(3) = 11 c) f(8) = 13.

Example 2: Find the domain and range of the function mentioned in Example 1.
Solution:
For the domain, x + 1 ≥ 0 ⇒ x ≥ 1. So the domain is [1, ∞).
The vertical shift of the function is 7. So the range is y ≥ 7 (or) it can be written as [7, ∞).
Answer: Domain = [1, ∞) and Range = [7, ∞).

Example 3: Graph the function mentioned in Example 1 by using the information from Example 1 and Example 2.
Solution:
From Example 1, we found the points to be (0, 9), (3, 11), and (8, 13).
From Example 2, we found the domain to be [1, ∞) and the range to be [7, ∞). We will compute the function for x = 1 as well as it is the starting point in the domain.
f(1) = 2√(1 + 1) + 7 = 2(0) + 7 = 7. Thus, the corresponding point is (1, 7). Now, we will plot all this information.
Answer: The given function is graphed.
FAQs on Square Root Function
What is Parent Square Root Function?
The parent square root function is f(x) = √x. This function may be translated/dilated/reflected and can transform to the form f(x) = a√(b(x  h)) + k.
What is the Formula of Square Root Function?
The formula for the square root function is f(x) = √x. It means the output of each input value is equal to the square root of the input value. For example, f(25) = √25 = 5. Note that all inputs and outputs of a square root function are always nonnegative.
What is the Derivative of Square Root Function?
The derivative of the square root function f(x) = √x is calculated by the power rule of differentiation, d(x^{n})/dx = nx^{n1}. By this rule, d(√x)/dx = d(x^{1/2})/dx = (1/2) x^{(1/2)  1 }= (1/2) x^{1/2 }= 1/(2√x). Thus, the derivative of √x is 1/(2√x).
How to Graph Square Root of x?
To graph the square root of x, just note that its inputs and outputs are all nonnegative and hence its graph lies in the first quadrant. Further, to get the clear shape of the graph, calculate some points on it, by taking some random numbers for x and computing corresponding yvalues for them.
What is Square Root Function Domain?
The square root function cannot be evaluated for negative inputs. Try putting √(2) in the calculator, it shows an error. Thus, the square root function f(x) = √x takes in only the nonnegative values and hence its domain is the set of all nonnegative real numbers, [0, ∞). Going forward, if the function is like f(x) = a√(b(x  h)) + k, then its domain is x ≥ h.
What is the Difference Between Cube Root Graph and Square Root Graph?
The cube root graph can take in any real number as input and produces any real number as output. But the square root function takes in and produces only the nonnegative real numbers.
What are the Asymptotes of Square Root Function?
There is no horizontal line or vertical line that can break the graph of square root function and hence it has no vertical/horizontal asymptotes.
What is the Integral of Square Root Function?
The integral of the square root function √x can be found using the power rule of integration ∫x^{n} dx = x^{n+1}/(n + 1) + C.. Using this, ∫√x dx = ∫x^{1/2} dx = x^{(1/2 + 1)}/(1/2 + 1) + C = x^{3/2}/(3/2) + C = (2/3) x^{3/2} + C.
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