5. \[$ G(x) = 3(2^x) \$\]$\[ \begin{array}{|c|c|} \hline x & Y \\ \hline -3 & 0.125 \\ \hline -2 & 0.25 \\ \hline -1 & 1.5 \\ \hline 0 & 1 \\ \hline 1 & 6 \\ \hline 2 & 12 \\ \hline 3 & 24 \\ \hline \end{array} \\]Asymptote: \[$ Y

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Understanding the Exponential Function g(x) = 3(2^x)

The exponential function is a fundamental concept in mathematics, and it plays a crucial role in various fields such as science, engineering, and economics. In this article, we will delve into the world of exponential functions, specifically the function g(x) = 3(2^x), and explore its properties, behavior, and applications.

What is an Exponential Function?

An exponential function is a mathematical function of the form f(x) = a^x, where 'a' is a positive constant and 'x' is the variable. The function is called exponential because the variable 'x' is raised to a power, and the result is multiplied by a constant 'a'. Exponential functions are characterized by their rapid growth or decay, depending on the value of 'a'.

The Function g(x) = 3(2^x)

The function g(x) = 3(2^x) is a specific type of exponential function, where 'a' is 2 and the constant multiplier is 3. This function can be rewritten as g(x) = 3 * 2^x, where 3 is the constant multiplier and 2^x is the exponential term.

Graphing the Function

To visualize the behavior of the function g(x) = 3(2^x), we can create a graph using the given data points. The graph will show the rapid growth of the function as 'x' increases.

x y
-3 0.125
-2 0.25
-1 1.5
0 1
1 6
2 12
3 24

Asymptote

The asymptote of an exponential function is the horizontal line that the function approaches as 'x' increases without bound. In the case of the function g(x) = 3(2^x), the asymptote is the x-axis, which is y = 0.

Properties of the Function

The function g(x) = 3(2^x) has several properties that are worth noting:

  • Domain: The domain of the function is all real numbers, denoted as (-∞, ∞).
  • Range: The range of the function is all positive real numbers, denoted as (0, ∞).
  • Periodicity: The function is not periodic, meaning it does not repeat itself at regular intervals.
  • Symmetry: The function is not symmetric about the y-axis, meaning it does not have a mirror image on the other side of the y-axis.

Behavior of the Function

The function g(x) = 3(2^x) exhibits rapid growth as 'x' increases. This is because the exponential term 2^x grows exponentially, and the constant multiplier 3 amplifies this growth. As 'x' approaches negative infinity, the function approaches 0, and as 'x' approaches positive infinity, the function approaches infinity.

Applications of the Function

Exponential functions like g(x) = 3(2^x) have numerous applications in various fields, including:

  • Population growth: Exponential functions can be used to model population growth, where the population grows at a rate proportional to the current population.
  • Finance: Exponential functions can be used to calculate compound interest, where the interest is added to the principal at regular intervals.
  • Science: Exponential functions can be used to model the growth of bacteria, the decay of radioactive materials, and the spread of diseases.

Conclusion

In conclusion, the function g(x) = 3(2^x) is a specific type of exponential function that exhibits rapid growth as 'x' increases. The function has several properties, including a domain of all real numbers, a range of all positive real numbers, and a non-periodic and non-symmetric behavior. The function has numerous applications in various fields, including population growth, finance, and science. By understanding the properties and behavior of exponential functions like g(x) = 3(2^x), we can better model and analyze real-world phenomena.

References

  • [1] "Exponential Functions" by Math Is Fun
  • [2] "Exponential Growth" by Khan Academy
  • [3] "Exponential Decay" by Wolfram MathWorld

Further Reading

For further reading on exponential functions, we recommend the following resources:

  • "Exponential Functions" by Wolfram MathWorld
  • "Exponential Growth and Decay" by Math Open Reference
  • "Exponential Functions in Real-World Applications" by ScienceDirect
    Q&A: Exponential Function g(x) = 3(2^x)

In this article, we will answer some frequently asked questions about the exponential function g(x) = 3(2^x).

Q: What is the domain of the function g(x) = 3(2^x)?

A: The domain of the function g(x) = 3(2^x) is all real numbers, denoted as (-∞, ∞).

Q: What is the range of the function g(x) = 3(2^x)?

A: The range of the function g(x) = 3(2^x) is all positive real numbers, denoted as (0, ∞).

Q: Is the function g(x) = 3(2^x) periodic?

A: No, the function g(x) = 3(2^x) is not periodic, meaning it does not repeat itself at regular intervals.

Q: Is the function g(x) = 3(2^x) symmetric about the y-axis?

A: No, the function g(x) = 3(2^x) is not symmetric about the y-axis, meaning it does not have a mirror image on the other side of the y-axis.

Q: What is the asymptote of the function g(x) = 3(2^x)?

A: The asymptote of the function g(x) = 3(2^x) is the x-axis, which is y = 0.

Q: How does the function g(x) = 3(2^x) behave as x approaches negative infinity?

A: As x approaches negative infinity, the function g(x) = 3(2^x) approaches 0.

Q: How does the function g(x) = 3(2^x) behave as x approaches positive infinity?

A: As x approaches positive infinity, the function g(x) = 3(2^x) approaches infinity.

Q: What are some real-world applications of the function g(x) = 3(2^x)?

A: Some real-world applications of the function g(x) = 3(2^x) include:

  • Modeling population growth
  • Calculating compound interest
  • Modeling the growth of bacteria
  • Modeling the decay of radioactive materials
  • Modeling the spread of diseases

Q: How can I graph the function g(x) = 3(2^x)?

A: You can graph the function g(x) = 3(2^x) using a graphing calculator or a computer algebra system. You can also use a table of values to create a graph.

Q: How can I find the value of the function g(x) = 3(2^x) at a specific value of x?

A: You can find the value of the function g(x) = 3(2^x) at a specific value of x by plugging the value of x into the function and evaluating it.

Q: What is the derivative of the function g(x) = 3(2^x)?

A: The derivative of the function g(x) = 3(2^x) is g'(x) = 3 * 2^x * ln(2), where ln(2) is the natural logarithm of 2.

Q: What is the integral of the function g(x) = 3(2^x)?

A: The integral of the function g(x) = 3(2^x) is ∫g(x) dx = 3 * (2^x) / ln(2) + C, where C is the constant of integration.

Conclusion

In this article, we have answered some frequently asked questions about the exponential function g(x) = 3(2^x). We hope that this article has been helpful in understanding the properties and behavior of this function. If you have any further questions, please don't hesitate to ask.