Which Is The Graph Of $g(x) = 2^{x-1} + 3$?

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Introduction

Graphing functions is an essential part of mathematics, and it's crucial to understand how to identify and graph various types of functions. In this article, we will focus on graphing the function $g(x) = 2^{x-1} + 3$. This function is an exponential function, and we will explore its properties and characteristics to determine its graph.

Understanding Exponential Functions

Exponential functions are a type of function that has the form $f(x) = a^x$, where $a$ is a positive real number. The graph of an exponential function is a curve that increases or decreases rapidly as $x$ increases. The base of the exponential function, $a$, determines the rate at which the function increases or decreases.

Properties of the Function $g(x) = 2^{x-1} + 3$

The function $g(x) = 2^{x-1} + 3$ is an exponential function with a base of 2. The exponent is $x-1$, which means that the function will increase or decrease as $x$ increases. The constant term 3 is added to the function, which shifts the graph of the function up by 3 units.

Graphing the Function

To graph the function $g(x) = 2^{x-1} + 3$, we can start by graphing the base function $f(x) = 2^x$. This function is an exponential function with a base of 2, and its graph is a curve that increases rapidly as $x$ increases.

Shifting the Graph

To graph the function $g(x) = 2^{x-1} + 3$, we need to shift the graph of the base function $f(x) = 2^x$ down by 1 unit. This is because the exponent is $x-1$, which means that the function is shifted down by 1 unit.

Adding the Constant Term

Finally, we need to add the constant term 3 to the graph of the function. This shifts the graph of the function up by 3 units.

Conclusion

In conclusion, the graph of the function $g(x) = 2^{x-1} + 3$ is a curve that increases rapidly as $x$ increases. The graph is shifted down by 1 unit due to the exponent $x-1$, and it is shifted up by 3 units due to the constant term 3.

Step-by-Step Solution

Here is a step-by-step solution to graph the function $g(x) = 2^{x-1} + 3$:

1. Graph the base function $f(x) = 2^x$.
2. Shift the graph of the base function down by 1 unit to get the graph of the function $f(x) = 2^{x-1}$.
3. Add the constant term 3 to the graph of the function to get the graph of the function $g(x) = 2^{x-1} + 3$.

Example

Here is an example of how to graph the function $g(x) = 2^{x-1} + 3$:

• Graph the base function $f(x) = 2^x$.
• Shift the graph of the base function down by 1 unit to get the graph of the function $f(x) = 2^{x-1}$.
• Add the constant term 3 to the graph of the function to get the graph of the function $g(x) = 2^{x-1} + 3$.

Graph

Here is the graph of the function $g(x) = 2^{x-1} + 3$:

[Insert graph here]

Final Answer

The final answer is the graph of the function $g(x) = 2^{x-1} + 3$.

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Introduction

In our previous article, we discussed the graph of the function $g(x) = 2^{x-1} + 3$. We explored the properties and characteristics of this function, and we provided a step-by-step solution to graph the function. In this article, we will answer some frequently asked questions about the graph of the function $g(x) = 2^{x-1} + 3$.

Q&A

Q: What is the base of the exponential function $g(x) = 2^{x-1} + 3$?

A: The base of the exponential function $g(x) = 2^{x-1} + 3$ is 2.

Q: What is the exponent of the exponential function $g(x) = 2^{x-1} + 3$?

A: The exponent of the exponential function $g(x) = 2^{x-1} + 3$ is $x-1$.

Q: How does the graph of the function $g(x) = 2^{x-1} + 3$ change as $x$ increases?

A: The graph of the function $g(x) = 2^{x-1} + 3$ increases rapidly as $x$ increases.

Q: What is the effect of the constant term 3 on the graph of the function $g(x) = 2^{x-1} + 3$?

A: The constant term 3 shifts the graph of the function $g(x) = 2^{x-1} + 3$ up by 3 units.

Q: How do you graph the function $g(x) = 2^{x-1} + 3$?

A: To graph the function $g(x) = 2^{x-1} + 3$, you need to graph the base function $f(x) = 2^x$, shift the graph of the base function down by 1 unit, and then add the constant term 3 to the graph.

Q: What is the final answer to graphing the function $g(x) = 2^{x-1} + 3$?

A: The final answer to graphing the function $g(x) = 2^{x-1} + 3$ is the graph of the function itself.

Additional Questions

Q: Can you provide more examples of graphing exponential functions?

A: Yes, we can provide more examples of graphing exponential functions. Here are a few examples:

• Graph the function $f(x) = 3^x$.
• Graph the function $f(x) = 4^{x-2}$.
• Graph the function $f(x) = 2^{x+1}$.

Q: How do you determine the domain and range of an exponential function?

A: To determine the domain and range of an exponential function, you need to consider the base and the exponent. The domain of an exponential function is all real numbers, and the range is all positive real numbers.

Q: Can you provide more information about the properties of exponential functions?

A: Yes, we can provide more information about the properties of exponential functions. Here are a few properties:

• The graph of an exponential function is a curve that increases or decreases rapidly as $x$ increases.
• The base of an exponential function determines the rate at which the function increases or decreases.
• The exponent of an exponential function determines the direction of the graph.

Conclusion

In conclusion, the graph of the function $g(x) = 2^{x-1} + 3$ is a curve that increases rapidly as $x$ increases. The graph is shifted down by 1 unit due to the exponent $x-1$, and it is shifted up by 3 units due to the constant term 3. We hope that this Q&A article has provided you with a better understanding of the graph of the function $g(x) = 2^{x-1} + 3$.