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
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 on finding the domain of a function and mathematics tutorials and problems.