Graph The Exponential Function G ( X ) = 4 X + 1 G(x) = 4^x + 1 G ( X ) = 4 X + 1 .1. Plot Two Points On The Graph Of The Function.2. Draw The Asymptote.3. Give The Domain And Range Of The Function Using Interval Notation.

Graphing the Exponential Function $g(x) = 4^x + 1$

The exponential function $g(x) = 4^x + 1$ is a type of mathematical function that exhibits rapid growth as the input value $x$ increases. In this article, we will explore the graph of this function, including plotting two points, drawing the asymptote, and determining the domain and range of the function using interval notation.

Plotting Two Points on the Graph of the Function

To plot two points on the graph of the function $g(x) = 4^x + 1$, we need to choose two values of $x$ and calculate the corresponding values of $g(x)$. Let's choose $x = 0$ and $x = 1$ as our two values.

For $x = 0$, we have:

$$g(0) = 4^0 + 1 = 1 + 1 = 2$$

So, the point $(0, 2)$ lies on the graph of the function.

For $x = 1$, we have:

$$g(1) = 4^1 + 1 = 4 + 1 = 5$$

So, the point $(1, 5)$ lies on the graph of the function.

Drawing the Asymptote

The asymptote of an exponential function is a horizontal line that the function approaches as $x$ increases without bound. In the case of the function $g(x) = 4^x + 1$, the asymptote is the line $y = 1$.

To draw the asymptote, we can use the fact that as $x$ increases without bound, the value of $4^x$ also increases without bound. Therefore, the value of $g(x) = 4^x + 1$ will approach $1$ as $x$ increases without bound.

Determining the Domain and Range of the Function

The domain of a function is the set of all possible input values $x$ for which the function is defined. In the case of the function $g(x) = 4^x + 1$, the domain is all real numbers, since $4^x$ is defined for all real numbers $x$.

The range of a function is the set of all possible output values $g(x)$ for which the function is defined. In the case of the function $g(x) = 4^x + 1$, the range is all real numbers greater than or equal to $1$, since $4^x$ is always positive and adding $1$ shifts the range up by $1$.

In interval notation, the domain and range of the function can be written as:

Domain: $(-\infty, \infty)$ Range: $[1, \infty)$

Graphing the Function

To graph the function $g(x) = 4^x + 1$, we can use the points $(0, 2)$ and $(1, 5)$ that we plotted earlier, as well as the asymptote $y = 1$.

The graph of the function will be a curve that approaches the asymptote $y = 1$ as $x$ increases without bound. The curve will also pass through the points $(0, 2)$ and $(1, 5)$.

Conclusion

In this article, we have explored the graph of the exponential function $g(x) = 4^x + 1$. We have plotted two points on the graph, drawn the asymptote, and determined the domain and range of the function using interval notation. The graph of the function is a curve that approaches the asymptote $y = 1$ as $x$ increases without bound, and passes through the points $(0, 2)$ and $(1, 5)$.

Key Takeaways

• The domain of the function $g(x) = 4^x + 1$ is all real numbers.
• The range of the function $g(x) = 4^x + 1$ is all real numbers greater than or equal to $1$.
• The graph of the function $g(x) = 4^x + 1$ approaches the asymptote $y = 1$ as $x$ increases without bound.
• The graph of the function $g(x) = 4^x + 1$ passes through the points $(0, 2)$ and $(1, 5)$.

Further Exploration

• Graph the function $g(x) = 4^x + 1$ using a graphing calculator or computer software.
• Explore the behavior of the function as $x$ approaches negative infinity.
• Determine the equation of the asymptote of the function $g(x) = 4^x + 1$.
• Graph the function $g(x) = 4^x + 1$ on a logarithmic scale.
  Q&A: Graphing the Exponential Function $g(x) = 4^x + 1$

In this article, we will answer some common questions about graphing the exponential function $g(x) = 4^x + 1$. Whether you are a student, teacher, or simply interested in mathematics, this article will provide you with a deeper understanding of the graph of this function.

Q: What is the domain of the function $g(x) = 4^x + 1$?

A: The domain of the function $g(x) = 4^x + 1$ is all real numbers, since $4^x$ is defined for all real numbers $x$.

Q: What is the range of the function $g(x) = 4^x + 1$?

A: The range of the function $g(x) = 4^x + 1$ is all real numbers greater than or equal to $1$, since $4^x$ is always positive and adding $1$ shifts the range up by $1$.

Q: How do I graph the function $g(x) = 4^x + 1$?

A: To graph the function $g(x) = 4^x + 1$, you can use the points $(0, 2)$ and $(1, 5)$ that we plotted earlier, as well as the asymptote $y = 1$. The graph of the function will be a curve that approaches the asymptote $y = 1$ as $x$ increases without bound.

Q: What is the asymptote of the function $g(x) = 4^x + 1$?

A: The asymptote of the function $g(x) = 4^x + 1$ is the line $y = 1$. This means that as $x$ increases without bound, the value of $g(x)$ will approach $1$.

Q: How do I determine the equation of the asymptote of the function $g(x) = 4^x + 1$?

A: To determine the equation of the asymptote of the function $g(x) = 4^x + 1$, you can use the fact that as $x$ increases without bound, the value of $4^x$ also increases without bound. Therefore, the value of $g(x) = 4^x + 1$ will approach $1$ as $x$ increases without bound.

Q: Can I graph the function $g(x) = 4^x + 1$ on a logarithmic scale?

A: Yes, you can graph the function $g(x) = 4^x + 1$ on a logarithmic scale. This will allow you to see the behavior of the function as $x$ approaches negative infinity.

Q: What are some common mistakes to avoid when graphing the function $g(x) = 4^x + 1$?

A: Some common mistakes to avoid when graphing the function $g(x) = 4^x + 1$ include:

• Not using the correct asymptote
• Not plotting the correct points
• Not using the correct scale
• Not considering the behavior of the function as $x$ approaches negative infinity

Q: How can I use the graph of the function $g(x) = 4^x + 1$ in real-world applications?

A: The graph of the function $g(x) = 4^x + 1$ can be used in a variety of real-world applications, including:

• Modeling population growth
• Modeling financial growth
• Modeling chemical reactions
• Modeling physical systems

Conclusion

In this article, we have answered some common questions about graphing the exponential function $g(x) = 4^x + 1$. Whether you are a student, teacher, or simply interested in mathematics, this article will provide you with a deeper understanding of the graph of this function.

Key Takeaways

• The domain of the function $g(x) = 4^x + 1$ is all real numbers.
• The range of the function $g(x) = 4^x + 1$ is all real numbers greater than or equal to $1$.
• The graph of the function $g(x) = 4^x + 1$ approaches the asymptote $y = 1$ as $x$ increases without bound.
• The graph of the function $g(x) = 4^x + 1$ passes through the points $(0, 2)$ and $(1, 5)$.

Further Exploration

• Graph the function $g(x) = 4^x + 1$ using a graphing calculator or computer software.
• Explore the behavior of the function as $x$ approaches negative infinity.
• Determine the equation of the asymptote of the function $g(x) = 4^x + 1$.
• Graph the function $g(x) = 4^x + 1$ on a logarithmic scale.