Find the range of square root functions; examples and matched problems with their answers at the bottom of the page.
Graphical Analysis of Range of Square Root Functions
The range of a function y = f(x) is the set of values y takes for all values of x within the domain of f.
What is the range of f(x) =
√x?
The domain of f above is the set of all values of x in the interval [0 , +∞)
As x takes values from 0 to +∞,
√x takes values from
√0 to
√+∞ (see blue graph below). Hence the range of
√x is given by the interval: [0 , +∞)
The range of function of the form
√x + k, (see red and green graphs below) is given by the interval:[0 , +∞). Right and left shifts do not affect the range of function.
We can also write the range y = f(x) =
√x + k of in the following inequality form
y ≥ 0 or
√x + k ≥ 0
Example 1:
Find the range of function f
defined by
f(x) =
√x  1
Solution to Example 1
We know, from the discussion above, that the range of function f(x) =
√x is given by the interval [0 , +∞).
The graph of the given function f(x) =
√x  1 is the graph of √ x shifted 1 unit to the right. A shift to the right does not affect the range. Hence the range of f(x) =
√x  1 is also given by the interval: [ 0 , +∞)
Matched Problem 1:
Find the range of
function f defined by
f(x) =
√x + 5
Example 2:
Find the range of function f
defined by
f(x) = 
√x + 2
Solution to Example 2
We first start with the range of values of expression
√x + 2 which may be written in inequality form as follows
√x + 2 ≥ 0
Multiply both sides of the inequality by 1 to obtain to obtain the inequality

√x + 2 ≤ 0
The range of the expression 
√x + 2 which is also the range of the given function is given by the interval ( ∞ , 0]
Matched Problem 2:
Find the range of
function f defined by
f(x) = 
√x  4
Example 3:
Find the range of function f
defined by
f(x) =  2
√x + 3 + 5
Solution to Example 3
The range of values of
√x + 3 may be written as an inequality
√x + 3 ≥ 0
Multiply both sides by 2 to obtain
 2
√x + 3 ≤ 0
Add 5t both sides of the above inequality to obtain
 2
√x + 3 + 5 ≤ 5
The range of values of the expression on the left side of the inequality, which is also the range of the given function, is given by the interval
(  ∞ , 5 ]
Matched Problem 3:
Find the range of
function f defined by
f(x) =  5
√x + 3  10
Example 4:
Find the range of function f
defined by
f(x) =
√16  x^{2}
Solution to Example 4
We first need to find the domain of the given function defined as the values of x such that
16  x^{2} ≥ 0
The solution set to the above inequality is the domain of f(x) and is given by the interval
[ 4 , 4 ]
The range of values of 16  x^{2} for x in the interval [ 4 , 4 ] (domain) is given by the interval [0 , 16] since the graph is a parabola with a maximum at the point (0 , 16). The given function is the square root of 16  x^{2} and therefore has the range defined by the interval [ √ 0 , √ 16 ] = [ 0 , 4 ]. See graphs below for better understanding
Matched Problem 4:
Find the range of
function f defined by
f(x) =
√4  x^{2}
Example 5:
Find the range of function f
defined by
f(x) =
√x^{2}  25
Solution to Example 5
The domain of the given function is the set of x values such that
x^{2}  25 ≥ 0
The solution set to the above inequality is the domain of f(x) and is given by the interval
(  ∞ , 5] ∪ [5 , + ∞)
For x in the interval (  ∞ , 5] ∪ [5 , + ∞) the range of the expression x^{2}  25 is given by the interval [ 0 , +∞). The given function is the square root of x^{2}  25. Hence the range of the given function is given by the interval [ √ 0 , √+∞) = [ 0 , +∞) . See graph below for better understanding.
Matched Problem 5:
Find the range of
function f defined by
f(x) =
√x^{2}  1
Example 6:
Find the range of function f
defined by
f(x) =
√x^{2}  4x + 8
Solution to Example 6
The domain of the given function is the set of x values such that
x^{2}  4x + 8 ≥ 0
The discriminant of the quadratic expression x^{2}  4x + 8 is given by
(4)^{2}  4 (1)(8) = 16
Since the discriminant is negative, the quadratic expression is either positive or negative for all values of x. A test with x = 0 reveals the expression x^{2}  4x + 8 is always positive and therefore the domain of the given function is the set of all real numbers.
We next find the range of the expression x^{2}  4x + 8 which may be written as
x^{2}  4x + 8 = (x  2)^{2} + 4
The graphs of (x  2)^{2} + 4 is a parabola with a minimum at (2 , 4) (the vertex). Hence the range of x^{2}  4x + 8 is given by the interval [ 4 , +∞). The given function is the square root of x^{2}  4x + 8 and therefore has the range given by [ √ 4 , √+∞ ) = [ 2 , +∞). See graph below for better understanding.