Find Range of Square Root Functions

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

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

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.

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.

Matched Problem 6:

Find the range of function f defined by
f(x) = x2 + 2x + 10

Answers to the Above Matched Problems

1)   [0 , +∞)
2)   ( -∞ , 0 ]
3)   ( -∞ , -10 ]
4)   [ 0 , 2 ]
5)   [ 0 , +∞)
6)   [ 3 , +∞)