Chapter 7.6, Problem 5STP

For Exercises $1-6,$ choose the correct letter.

Which of the following statements is not true regarding the function $y=2^x$?

A. The function is an exponential function.

B. The function has a domain of all real numbers.

C. As the value of $x$ gets very large, the value of $y$ gets close to zero.

D. As the value of $x$ increases by one, the value of $y$ doubles.

To determine

The statement that is not true regarding the function $y=2^x$ .

Answer to Problem 5STP

The option C is not true.

Explanation of Solution

Given information:

The given statements are

A. The function is an exponential function.

B. The function has a domain of all real numbers.

C. As the value of x gets very large, the value of y gets close to zero.

D. As the value of x increases by one, the value of y doubles.

Concept used:

Definition: A function of the form $y=b^x$ is an exponential function.

The function $y=2^x$ by definition is an exponential function, so option A is true

The function $y=2^x$ is defined for all real numbers its domain is set of all real numbers, so option B is true.

The function $y=2^x$ is an increasing exponential function, therefore, as the value of x gets very large, the value of y gets large, so option C is not true.

Let $y_0(x)=2^x$ ,

  $\Rightarrow y_1(x+1)=2^{x+1}=2\cdot 2^x=2\cdot y_0$

  $\Rightarrow y_2(x+1+1)=2^{x+1+!}=2\cdot 2^{x+1}=2\cdot y_1$ …and so on.

Thus, as the value of x increases by one, the value of y doubles, so option D is true.

Conclusion:

The option C is not true.