# 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 \[ f(x) = e^{-x+2} \]__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
\[ y = - 2 e^{-x^2} + 3\]
Solve the above for \( x \)

\( y - 3 = -2 e^{-x^2} \)

\( e^{-x^2} = \frac{y - 3}{-2} \)

\( -x^2 = \ln \left[ \frac{y - 3}{-2} \right] \)

\( x = \pm \sqrt{ - \ln \left[ \frac{y - 3}{-2} \right] } \)

\( x \) is real if the argument of \( \ln \) is positive and the radicand is positive or zero. Hence the following inequalities

\( \frac{y - 3}{-2} > 0 \) and \( - \ln \left[ \frac{y - 3}{-2} \right] \geq 0 \)

the solution set of \( \frac{y - 3}{-2} > 0 \) is \( y \lt 3 \)

the solution set of \( - \ln \left[ \frac{y - 3}{-2} \right] \geq 0 \) is given by \( \frac{y - 3}{-2} \leq 1 \) which gives \( y \geq 1 \)

the range of \( f \) is given by

\( 1 \leq y \lt 3 \) 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

\[ y = 4 e^{-x^2} -7\]## Answers to Matched Problems

- \( (0 , +\infty) \)
- \( (-\infty , -2) \)
- \( (-\infty , -6] \)
- \( (-7 , -3] \)

## More References and Links

Find the domain of a functionmathematics tutorials and problems .