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 Solutions

Example 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
x2 = 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)
-x2 = 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

1. (0 , +∞)
2. (-∞ , -2)
3. (-∞ , -6]
4. (-7 , -3]