To determine whether the function [tex]\( f(x) = 10 \cdot \left(\frac{1}{15}\right)^x \)[/tex] represents growth or decay, we need to look at the base of the exponent, which is [tex]\( \frac{1}{15} \)[/tex].
An exponential function can be written in the form [tex]\( f(x) = a \cdot b^x \)[/tex], where:
- [tex]\( a \)[/tex] is the initial value (in this case, 10)
- [tex]\( b \)[/tex] is the base of the exponential function
The behavior of the exponential function depends on the value of the base [tex]\( b \)[/tex]:
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
Here, the base [tex]\( b \)[/tex] is [tex]\( \frac{1}{15} \)[/tex]. This is a fraction that lies between 0 and 1.
Since [tex]\( 0 < \frac{1}{15} < 1 \)[/tex], the function [tex]\( f(x) = 10 \cdot \left(\frac{1}{15}\right)^x \)[/tex] represents exponential decay.
Therefore, the correct statement is:
"The function represents exponential decay because the base equals [tex]\( \frac{1}{15} \)[/tex]."
