To compute the domain and range of an exponential decay function, you need to consider the general form of the function and its behavior. Exponential decay functions generally follow the form:

$f(x) = a \cdot b^x$

Where:

• $a$ is a constant that affects the initial value.
• $b$ is the base of the exponential, where $0 < b < 1$ for decay.
• $x$ is the input or independent variable.

1. Domain of an Exponential Decay Function

The domain is the set of all possible input values (i.e., values of $x$) for which the function is defined.

For any exponential function $f(x) = a \cdot b^x$, the domain is all real numbers:

• There are no restrictions on the value of $x$, because you can plug any real number into the exponent and still get a valid result.

Domain:
$\boxed{(-\infty, \infty)}$

2. Range of an Exponential Decay Function

The range is the set of all possible output values (i.e., values of $f(x)$) for the given function.

For exponential decay:

• As $x \to \infty$, $b^x$ approaches 0, so $f(x) \to 0$.
• As $x \to -\infty$, $b^x \to \infty$ (since $0 < b < 1$).
• The function will always be positive (assuming $a > 0$) because the exponential term never crosses the x-axis, and $a$ is a constant multiplier.

Therefore, the range of an exponential decay function is positive values starting from 0 but never reaching it.

Range:
$\boxed{(0, \infty)}$

Example:

For $f(x) = 5 \cdot (0.5)^x$:

• The domain is all real numbers: $(-\infty, \infty)$.
• The range is all positive numbers: $(0, \infty)$, since the function decays toward 0 but never actually reaches 0.

---

Summary:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Would you like more details or examples on specific functions? Feel free to ask!

Here are some related questions:

1. What happens to the range if $a$ is negative in an exponential decay function?
2. How do you determine the horizontal asymptote of an exponential decay function?
3. Can exponential decay functions be defined for complex numbers?
4. What is the effect of changing the base $b$ to a value larger than 1 in an exponential function?
5. How do you graph an exponential decay function?

Tip: Always remember that for exponential decay, the base $b$ must be between 0 and 1 for the function to decay (i.e., get smaller over time).