Chapter 7.7, Problem 29PPE

To determine

Tostate:Whether the given equation represents exponential growth, exponential decay or neither.

Answer to Problem 29PPE

The function $y\left(x\right)=2\cdot {0.68}^x$ represents exponential decay.

Explanation of Solution

Given information:

The given exponential function is $y\left(x\right)=2\cdot {0.68}^x$

Conceptused:

A function of the form $y\left(x\right)=a\cdot b^x$ is known as the exponential function. As $y\left(0\right)=a\cdot b^0=a$ , so “ a ” is called as initial amount of function $y\left(x\right)=a\cdot b^x$ . If the factor $b>1$ then the function $y\left(x\right)=a\cdot b^x$ represents exponential growth, and if the factor $b<1$ then the function $y\left(x\right)=a\cdot b^x$ represents exponential decay.

The given exponential function is

  $y\left(x\right)=2\cdot {\left(0.68\right)}^x$  ... (1)

Comparing given function (1) with the function $y\left(x\right)=a\cdot b^x$ , we have $b=0.68<1$ .

Hence, the function $y\left(x\right)=2\cdot {\left(0.68\right)}^x$ represents exponential decay.

Conclusion:

The function $y\left(x\right)=2\cdot {0.68}^x$ represents exponential decay.