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

$y = 4 e^{-x^2} -7$

1. $$(0 , +\infty)$$
2. $$(-\infty , -2)$$
3. $$(-\infty , -6]$$
4. $$(-7 , -3]$$