Question

which statements are true about exponential decay functions? check all that apply.

the domain is all real numbers.
as the input increases, the output increases.
the graph is the same as that of an exponential growth function.
the base must be less than 1 and greater than 0.
the function has a constant multiplicative rate of change.

Explanation:

Analyze the definition and domain of exponential decay

$$ f(x) = a \cdot b^x \quad (a > 0, \ 0 < b < 1) $$

$$ \text{Domain} = (-\infty, \infty) $$

Evaluate the behavior and rate of change

$$ 0 < b < 1 \implies \text{As } x \text{ increases, } f(x) \text{ decreases} $$

$$ \frac{f(x+1)}{f(x)} = b \quad (\text{constant multiplicative rate of change}) $$

Compare with exponential growth

$$ \text{Growth base } b > 1 eq \text{Decay base } 0 < b < 1 $$

Answer:

• The domain is all real numbers. (Correct answer)
• As the input increases, the output increases.
• The graph is the same as that of an exponential growth function.
• The base must be less than 1 and greater than 0. (Correct answer)
• The function has a constant multiplicative rate of change. (Correct answer)