Square Root Function - Graph, Domain, Range, Examples - Cuemath

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 y2 = x. i.e., if y2 = x ⇒ y = √x. i.e., if 'x' is the square of 'y' then 'y' is the square root of 'x'. Some examples are:

• 22 = 4 ⇒ √4 = 2
• 42 = 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.

1.	What is Square Root Function?
2.	Domain and Range of Square Root Function
3.	Square Root Graph
4.	Graphing Any Square Root Function
5.	Properties of Square Root Function
6.	FAQs on 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) = x2 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 one-one, it is a bijection and so it has an inverse. The graphs square root function f(x) = √x and its inverse g(x) = x2 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 non-negative 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 52 = 25 and (-5)2 = 25. But the range of the square root function (i.e., its y-values) 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 non-negative 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 x-intercept and y-intercept 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.