Find Range of Exponential Functions
Find the range of real valued exponential functions using different techniques. Matched problems are also included with their answers at the bottom of the page.

Examples with SolutionsExample 1
Find the Range of function f
defined by
Solution to Example 1

Let us first write the above function as an equation as follows

solve the above function for x
x + 2 = ln (y)
x = 2  ln (y)

x is a real number if y > 0 (argument of ln y must be positive). Hence the range of function f is given by
y > 0 or the interval (0 , +∞)
See graph of f below and examine the range graphically.
Matched Problem 1:
Find the range of
function f defined by
Example 2
Find the Range of function f
defined by
Solution to Example 2

Write the given function as an equation

Solve the above equation for x
x = (1 / 2)(ln (y  3) 1)

x is a real number for y  3 > 0 (argument of ln (y  3) must be positive). The range of the given function is then given by
y > 3 or in interval form (3 , +∞)
See graph of f below and examine the range graphically.
Matched Problem 2:
Find the range of
function f defined by
Example 3
Find the Range of function f
defined by
Solution to Example 3

Write the given function as an equation

Solve the above for x to obtain
x^{2} = ln(y  1)
x = + or  √[ ln(y  1) ]

The above solutions are real if
ln(y  1) ≥ 0
y  1 ≥ 1
y ≥ 2

Hence the range of the given function is given by
y ≥ 2 or in interval form [ 2 , + ∞ )
See graph of f below and examine the range graphically.
Matched Problem 3:
Find the range of
function f defined by
Example 4
Find the Range of function f
defined by
Solution to Example 4

We first write the given function as an equation as follows

Solve the above for x
y  3 = 2 e^{x2}
e^{x2} = (y  3) / (2)
x^{2} = ln [ (y  3) / (2) ]
x = + or  √(  ln [ (y  3) / (2) ])

x is real if the argument of ln is positive and the radicand is positive or zero. Hence the following inequalities
(y  3) / (2) > 0 and  ln [ (y  3) / (2) ] ≥ 0
the solution set of (y  3) / (2) > 0 is y < 3
the solution set of  ln [ (y  3) / (2) ] ≥ 0 is given by (y  3) / (2) ≤ 1 which gives y ≥ 1

the range of f is given by
1 ≤ y < 3 or in interval form [ 1 , 3 )
See graph of f below and examine the range graphically.
Matched Problem 4:
Find the range of
function f defined by
Answers to Matched Problems
1. (0 , +∞)
2. (∞ , 2)
3. (∞ , 6]
4. (7 , 3]
More References and LinksFind the domain of a function mathematics tutorials and problems.
