Question

which statements are true about exponential decay functions? choose three correct answers.
the domain is all real numbers.
the graph is the same as that of an exponential growth function.
the function has a constant multiplicative rate of change.
as the input increases, the output increases.
the base must be less than 1 and greater than 0.

Explanation:

Step1: Analyze "The domain is all real numbers"

Exponential functions (decay or growth) have the form \( y = ab^x \), where \( a
eq0 \), \( b>0 \), \( b
eq1 \). For any real number \( x \), \( b^x \) is defined (since \( b>0 \)), so the domain is all real numbers. This statement is true.

Step2: Analyze "The graph is the same as that of an exponential growth function"

Exponential growth functions have \( b > 1 \), so their graphs increase as \( x \) increases. Exponential decay functions have \( 0 < b < 1 \), so their graphs decrease as \( x \) increases. The shapes are different (one increasing, one decreasing), so this statement is false.

Step3: Analyze "The function has a constant multiplicative rate of change"

Exponential functions (decay or growth) have a constant multiplicative rate of change. For example, if \( y = ab^x \), then \( \frac{y(x + 1)}{y(x)}=\frac{ab^{x + 1}}{ab^x}=b \), which is constant. So this statement is true.

Step4: Analyze "As the input increases, the output increases"

For exponential decay (\( 0 < b < 1 \)), as \( x \) (input) increases, \( b^x \) decreases (since \( b \) is between 0 and 1). So the output \( y = ab^x \) (with \( a>0 \)) decreases as input increases. This statement is false.

Step5: Analyze "The base must be less than 1 and greater than 0"

The general form of an exponential decay function is \( y = ab^x \) where \( 0 < b < 1 \) (and \( a>0 \)). If \( b \geq 1 \) or \( b \leq 0 \), it's not a decay function. So this statement is true.

Answer:

• The domain is all real numbers.
• The function has a constant multiplicative rate of change.
• The base must be less than 1 and greater than 0.