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
Fig1. - Examples of Square Root Functions.
Examples with Solutions
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 - x2
Solution to Example 4
We first need to find the domain of the given function defined as the values of x such that
16 - x2 ≥ 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 - x2 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 - x2 and therefore has the range defined by the interval [ √ 0 , √ 16 ] = [ 0 , 4 ]. See graphs below for better understanding
Fig2. - Range of Square Root Functions.
Matched Problem 4:
Find the range of
function f defined by
f(x) =
√4 - x2
Example 5
Find the range of function f
defined by
f(x) =
√x2 - 25
Solution to Example 5
The domain of the given function is the set of x values such that
x2 - 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 x2 - 25 is given by the interval [ 0 , +∞). The given function is the square root of x2 - 25. Hence the range of the given function is given by the interval [ √ 0 , √+∞) = [ 0 , +∞) . See graph below for better understanding.
Fig3. - Range of Square Root Functions.
Matched Problem 5:
Find the range of
function f defined by
f(x) =
√x2 - 1
Example 6
Find the range of function f
defined by
f(x) =
√x2 - 4x + 8
Solution to Example 6
The domain of the given function is the set of x values such that
x2 - 4x + 8 ≥ 0
The discriminant of the quadratic expression x2 - 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 x2 - 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 x2 - 4x + 8 which may be written as
x2 - 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 x2 - 4x + 8 is given by the interval [ 4 , +∞). The given function is the square root of x2 - 4x + 8 and therefore has the range given by [ √ 4 , √+∞ ) = [ 2 , +∞). See graph below for better understanding.
