Question

How do you determine if the equation $y = - 5{\left( {\dfrac{1}{3}} \right)^{ - x}}$ represents exponential growth or decay?

Answer 1

Hint: The exponential function is the function in the given form:
 $y(x) = a.{b^{cx}}$
Here, $a,b$ and $c$ are constant, $a,c \ne 0$ and $b \ne 1$ .
As there are two types of exponential behaviors.
Exponential growth: the value of $y(x)$ tends to $\infty $ when $c \to \infty $ .
Exponential decay: it is inverse to exponential growth.

Complete step by step solution:
As exponentials function is in the form,
$y(x) = a.{b^{cx}}$
Here, $a,b$ and $c$ are constant, $a,c \ne 0$ and $b \ne 1$ .
So, comparing it with exponential function. We get,
 $a = - 5$ , $b = \left( {\dfrac{1}{3}} \right)$ and $c = - 1$
As there is general rule that we know is:
If $b > 1$ , then:
If $a,c$ are both positive or negative, we find exponential growth.
If $a,c$ have different signs, we find exponential decay.
If $b < 1$ , then:
If $a,c$ are both positive or negative, we find exponential decay.
If $a,c$ have different signs, we find exponential growth.

Therefore, by doing observation. We get,
 $\dfrac{1}{3} < 1$ and $a,c$ have same sign
So, from above we can conclude that $y = - 5{\left( {\dfrac{1}{3}} \right)^{ - x}}$ equation represents exponential decay.

Note: we shall keep in mind that $b$ is taken as a positive number, given that if take $b$ is negative, then some solution of the function will be a complex function, and we just take in account the real function.