To determine which function has the greatest decay rate, we need to analyze the base of the exponential functions. In general, an exponential decay function is written in the form:

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

where $a$ is a constant and $0 < b < 1$ determines the decay rate. The smaller the value of $b$, the faster the decay.

Let's compare the bases from the functions provided in the image:

1. $f(x) = 4 \cdot 0.75^x$ → Base is $0.75$
2. $f(x) = 8 \cdot 0.95^x$ → Base is $0.95$
3. $f(x) = 2 \cdot 0.82^x$ → Base is $0.82$
4. $f(x) = 6 \cdot 0.94^x$ → Base is $0.94$

Conclusion:

The function with the smallest base has the greatest decay rate. Here, the base 0.75 is the smallest, so the function $f(x) = 4 \cdot 0.75^x$ has the greatest decay rate.

Would you like more details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:

1. What would happen if the base of the exponential function was greater than 1?
2. How does the initial value (the constant before the base) affect the graph of the function?
3. Can you identify any real-world examples that follow exponential decay?
4. How would the decay rate change if the base was closer to 1?
5. How do we distinguish between exponential growth and decay just by looking at the equation?

Tip: The base of an exponential function directly influences whether it's a growth or decay process. A base less than 1 indicates decay, while a base greater than 1 represents growth.