Question

select the best description of the following exponential equation.
$f(x)=19(0.83)^x$
show your work here
hint: to add an exponent ($x^n$), type \exponent\ or press \^\
decaying by 83% with every unit of $x$.
growing by 83% with every unit of $x$.
decaying by 17% with every unit of $x$.
growing by 17% with every unit of $x$.

Explanation:

Step1: Recall exponential model

The general form of an exponential function is \( f(x) = a(b)^x \), where \( a \) is the initial amount, and \( b \) is the base. If \( 0 < b < 1 \), it's a decay function; if \( b > 1 \), it's a growth function.

Step2: Analyze the base

In \( f(x) = 19(0.83)^x \), the base \( b = 0.83 \). Since \( 0.83 < 1 \), it's a decay function.

Step3: Calculate decay rate

The decay rate \( r \) is found by \( r = 1 - b \). So \( r = 1 - 0.83 = 0.17 \), which is \( 17\% \). This means the function decays by \( 17\% \) for each unit increase in \( x \).

Answer:

Decaying by 17% with every unit of \( x \) (the option corresponding to this description, e.g., if the options are labeled as above, the correct one would be the option stating "Decaying by 17% with every unit of \( x \)")