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
\[ f(x) = e^{-x+2} \]

Solution to Example 1

Let us first write the above function as an equation as follows
\[ y = e^{-x+2} \]
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 \( (0 , +\infty) \)
See graph of \( f \) below and examine the range graphically.

Matched Problem 1:

Find the range of function \( f \) defined by
\[ f(x) = e^{-3x-2} \]

Example 2

Find the Range of function \( f \) defined by
\[ f(x) = e^{2x+1} + 3\]

Solution to Example 2

Write the given function as an equation
\[ y = e^{2x+1} + 3\]
Solve the above equation for \( x \)
\( x = \frac{1}{2}(\ln (y - 3) -1) \)
\( x \) 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 \( (3 , +\infty) \)

See graph of \( f \) below and examine the range graphically.

Matched Problem 2:

Find the range of function \( f \) defined by

\[ f(x) = - e^{-3x-2} - 2\]

Example 3

Find the Range of function \( f \) defined by
\[ f(x) = e^{x^2} + 1 \]

Solution to Example 3

Write the given function as an equation

\[ y = e^{x^2} + 1 \]
Solve the above for \( x \) to obtain
\( x^2 = \ln(y - 1) \)
\( x = \pm \sqrt{ \ln(y - 1) } \)
The above solutions are real if
\( \ln(y - 1) \geq 0 \)
\( y - 1 \geq 1 \)
\( y \geq 2 \)
Hence the range of the given function is given by
\( y \geq 2 \) or in interval form \( [ 2 , +\infty ) \)
See graph of \( f \) below and examine the range graphically.

Matched Problem 3:

Find the range of function \( f \) defined by

\[ f(x) = -e^{x^2} - 5\]

Example 4

Find the Range of function \( f \) defined by
\[ f(x) = - 2 e^{-x^2} + 3\]

Solution to Example 4

We first write the given function as an equation as follows
\[ y = - 2 e^{-x^2} + 3\]