In what follows, SQRT means square root.
The domain of function f defined by f(x) = SQRT ( x ) is the set of all real positive numbers and zero because the square root of negative numbers are not real numbers (think of SQRT ( 4), is it real?). In inequality form, the domain of f(x) = SQRT ( x ) is written as
x ≥ 0
in interval form the domain is given by
[ 0 , + infinity)
Example 1: Graph
f( x ) = SQRT (x)
and find the range of f.
Solution to Example 1:
Because the domain of f is the set of all positive real numbers and zero, we might construct a table of values as follows:
x  0  1  4  9  16 
SQRT (x)  0  1  2 
3 
4 
The values of x were selected so that the square root of these values are whole numbers which make it easy to plot the points shown in the table.
The range of f is given by the interval [0 , +infinity).
Example 2: Graph
f( x ) = SQRT (x  3)
and find the range of f.
Solution to Example 2:
First find the domain of the square root function given above by stating that the expression under the square root must be positive or equal to zero
x  3 ≥ 0
Solve the above inequality to obtain the domain of f as the set of all real values such that
x ≥ 3
We now select values of x in the domain to construct a table of values.
x  3  4  7  12 
SQRT (x  3)  0  1  2 
3 
The interval [0 , +infinity) represents the range of f.
Example 3: Graph
f( x ) =  SQRT ( 2x + 4) + 1
and find the range of f.
Solution to Example 3:
The domain of the function given above is found by setting
 2x + 4 ≥ 0
Solve the above inequality to obtain the domain of f as the set of all real values such that
x ≤ 2
We now select values of x in the domain of f to construct a table of values. These values are selected so that the square root term is a whole number and give points that are easy to plot.
x  2  3/2  0  5/2  6 
 SQRT (2 x + 4 ) + 1  1  0  1 
2 
3 
The range of f is given by the interval (infinity , 1].
Example 4: Graph
f( x ) = SQRT ( x^{ 2} + 4)
and find the range of f.
Solution to Example 4:
The domain of function given above is found by solving the polynomial inequality
 x^{ 2} + 4 ≥ 0
The solution set of the above inequality is given by the interval
[2 , 2]
which is also the domain of the above function.
Let us write the given function as an equation as follows
y = SQRT ( x^{ 2} + 4)
Square both sides and arrange to obtain.
x^{ 2} + y^{ 2} = 2^{ 2}
The equation obtained is that of a circle. Hence the graph of f(x) = SQRT ( x^{ 2} + 4) is the upper half of a circle sinsce SQRT ( x^{ 2} + 4) is positive. Hence the graph below.
The interval [0 , 2] represents the range of f.
Example 5: Graph
f( x ) = SQRT (x^{ 2}  9)
and find the range of f.
Solution to Example 5:
The domain of the function given above is found by solving
x^{ 2}  9 ≥ 0
Which gives a domain reprsented by
(infinity , 3] U [3 , + infinity)
We now select values of x in the domain of f to construct a table of values, noting f(x) = f(x) hence a symmetry of the graph with respect to the y axis.
x  3  5  8 
SQRT (x^{ 2}  9)  0  4  7.4 
The range of f is given by the interval [0 , + infinity).
Example 6: Graph
f( x ) = SQRT (x^{ 2}  6x + 9)
and find the range of f.
Solution to Example 6:
Let us use write the expression under the square root as a square as follows
x^{ 2}  6x + 9 = (x  3)^{ 2}
Hence
f( x ) = SQRT (x^{ 2}  6x + 9)
= SQRT ( (x  3)^{ 2} ) =  x  3 
The given function has been rewitten as an absolute value function. Function f may be written as a piecewise function and graphed as follows.
The range of f is given by the interval [0 , + infinity).
Example 7: Graph
f( x ) = SQRT (x^{ 2} + 4x + 6)
and find the range of f.
Solution to Example 7:
Use completing the square to rewtite the expression under the square root as follows
x^{ 2} + 4x + 6 = (x + 2)^{ 2} + 2
The expression under the square root is always positive hence the domain of f is the set of all real numbers. Let us first look at the graph of (x + 2)^{ 2} + 2. It is a parabola.
We would expect the graph of f to have the same axis of symmetry, the vertical line, x = 2 as the above graph. The table of values may constructed as follows.
x  2  0  2  4 
SQRT ( (x + 2)^{ 2} + 2 )  1.4  2.4  4.2 
6.2 
The range of f is given by the interval [SQRT(2) , + infinity).
More references and links on graphing.
Graphing Functions
