How To Find The Range Of An Exponential Function

Detect Range of Exponential Functions

Observe the range of real valued exponential functions using unlike techniques. Matched problems are too included with their answers at the bottom of the page.

Examples with Solutions

Example 1

Detect the Range of part f divers by

Solution to Case 1

Matched Problem one:

Find the range of function f defined past

Example 2

Find the Range of function f divers past

Solution to Example two

Matched Problem 2:

Find the range of part f defined past

Example 3

Find the Range of function f defined by

Solution to Example three

Write the given office as an equation
Solve the above for x to obtain
x² = ln(y - ane)
10 = + or - √[ ln(y - 1) ]
The to a higher place solutions are real if
ln(y - 1) ≥ 0
y - 1 ≥ ane
y ≥ two
Hence the range of the given function is given by
y ≥ 2 or in interval form [ 2 , + ∞ )

See graph of f beneath and examine the range graphically.

Matched Problem three:

Notice the range of function f defined by

Example iv

Find the Range of function f defined by

Solution to Case 4

We first write the given part as an equation as follows
Solve the in a higher place for x
y - 3 = -2 eastward^-10ⁱⁱ
due east^-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) / (-two) > 0 and - ln [ (y - three) / (-2) ] ≥ 0
the solution set of (y - 3) / (-2) > 0 is y < three
the solution gear up of - ln [ (y - 3) / (-2) ] ≥ 0 is given past (y - 3) / (-2) ≤ 1 which gives y ≥ 1
the range of f is given by
1 ≤ y < 3 or in interval form [ ane , 3 )