Consider The Graph Of The Function $f(x)=2^x$.Which Statements Describe Key Features Of The Function $g$ If $g(x)=f(x+2$\]?- Domain Of $\{x \mid -\infty \ \textless \ X \ \textless \ \infty\}$- Horizontal

Feb 25, 2025 by ADMIN 209 views

Understanding the Transformation of the Function $f(x)=2^x$

The function $f(x)=2^x$ is a fundamental exponential function that exhibits a rapid growth rate as the input value $x$ increases. In this article, we will explore the transformation of this function to obtain a new function $g(x)$ , and identify the key features of $g(x)$ .

The Transformation of $f(x)$ to $g(x)$

To obtain the function $g(x)$ , we apply a horizontal shift of 2 units to the right to the function $f(x)$ . This is achieved by replacing $x$ with $(x+2)$ in the function $f(x)$ . Therefore, the function $g(x)$ is given by:

g(x)=f(x+2)=2^{x+2}

Domain of $g(x)$

The domain of a function is the set of all possible input values for which the function is defined. Since the function $g(x)$ is an exponential function, it is defined for all real numbers. Therefore, the domain of $g(x)$ is:

\{x \mid -\infty \ \textless \ x \ \textless \ \infty\}

This means that $g(x)$ is defined for all real numbers, and there are no restrictions on the input values.

Horizontal Shift

As mentioned earlier, the function $g(x)$ is obtained by applying a horizontal shift of 2 units to the right to the function $f(x)$ . This means that the graph of $g(x)$ is identical to the graph of $f(x)$ , but shifted 2 units to the right.

Vertical Shift

In addition to the horizontal shift, the function $g(x)$ also undergoes a vertical shift. Since $g(x)=2^{x+2}$ , we can rewrite this as:

g(x)=2^2 \cdot 2^x

This shows that the graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of $2^2=4$ .

Key Features of $g(x)$

Based on the transformation of $f(x)$ to $g(x)$ , we can identify the following key features of $g(x)$ :

Domain: The domain of $g(x)$ is the same as the domain of $f(x)$ , which is all real numbers.
Horizontal Shift: The graph of $g(x)$ is identical to the graph of $f(x)$ , but shifted 2 units to the right.
Vertical Shift: The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 4.
Exponential Growth: The function $g(x)$ exhibits exponential growth, just like the function $f(x)$ .

Conclusion

In this article, we have explored the transformation of the function $f(x)=2^x$ to obtain a new function $g(x)$ . We have identified the key features of $g(x)$ , including its domain, horizontal shift, vertical shift, and exponential growth. These features are essential in understanding the behavior of the function $g(x)$ and its applications in various fields.

Key Takeaways

The function $g(x)$ is obtained by applying a horizontal shift of 2 units to the right to the function $f(x)$ .
The graph of $g(x)$ is identical to the graph of $f(x)$ , but shifted 2 units to the right.
The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 4.
The function $g(x)$ exhibits exponential growth, just like the function $f(x)$ .

Further Reading

For further reading on the topic of exponential functions and their transformations, we recommend the following resources:

By understanding the transformation of the function $f(x)=2^x$ to obtain a new function $g(x)$ , we can gain insights into the behavior of exponential functions and their applications in various fields.
Q&A: Understanding the Transformation of the Function $f(x)=2^x$

In our previous article, we explored the transformation of the function $f(x)=2^x$ to obtain a new function $g(x)$ . We identified the key features of $g(x)$ , including its domain, horizontal shift, vertical shift, and exponential growth. In this article, we will answer some frequently asked questions about the transformation of $f(x)$ to $g(x)$ .

Q: What is the domain of $g(x)$ ?

A: The domain of $g(x)$ is the same as the domain of $f(x)$ , which is all real numbers. This means that $g(x)$ is defined for all real numbers, and there are no restrictions on the input values.

Q: How does the graph of $g(x)$ differ from the graph of $f(x)$ ?

A: The graph of $g(x)$ is identical to the graph of $f(x)$ , but shifted 2 units to the right. This means that the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ by 2 units to the right.

Q: What is the effect of the vertical shift on the graph of $g(x)$ ?

A: The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 4. This means that the graph of $g(x)$ is 4 times taller than the graph of $f(x)$ .

Q: How does the exponential growth of $g(x)$ compare to the exponential growth of $f(x)$ ?

A: The function $g(x)$ exhibits exponential growth, just like the function $f(x)$ . However, the rate of growth of $g(x)$ is 4 times faster than the rate of growth of $f(x)$ .

Q: Can I use the transformation of $f(x)$ to $g(x)$ to solve other problems?

A: Yes, the transformation of $f(x)$ to $g(x)$ can be used to solve other problems involving exponential functions. By applying the same transformation to other exponential functions, you can obtain new functions with different properties.

Q: How can I visualize the transformation of $f(x)$ to $g(x)$ ?

A: You can visualize the transformation of $f(x)$ to $g(x)$ by plotting the graphs of both functions on the same coordinate plane. The graph of $g(x)$ will be identical to the graph of $f(x)$ , but shifted 2 units to the right.

Q: Can I use technology to help me visualize the transformation of $f(x)$ to $g(x)$ ?

A: Yes, you can use technology such as graphing calculators or computer software to help you visualize the transformation of $f(x)$ to $g(x)$ . These tools can help you plot the graphs of both functions and see the effect of the transformation.

Q: How can I apply the transformation of $f(x)$ to $g(x)$ to real-world problems?

A: The transformation of $f(x)$ to $g(x)$ can be applied to real-world problems involving exponential growth and decay. For example, you can use this transformation to model the growth of a population or the decay of a substance over time.

Conclusion

In this article, we have answered some frequently asked questions about the transformation of the function $f(x)=2^x$ to obtain a new function $g(x)$ . We have discussed the domain, horizontal shift, vertical shift, and exponential growth of $g(x)$ , and provided examples of how to apply the transformation to real-world problems. By understanding the transformation of $f(x)$ to $g(x)$ , you can gain insights into the behavior of exponential functions and their applications in various fields.

Key Takeaways

The domain of $g(x)$ is the same as the domain of $f(x)$ , which is all real numbers.
The graph of $g(x)$ is identical to the graph of $f(x)$ , but shifted 2 units to the right.
The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 4.
The function $g(x)$ exhibits exponential growth, just like the function $f(x)$ .
The transformation of $f(x)$ to $g(x)$ can be applied to real-world problems involving exponential growth and decay.

Further Reading

For further reading on the topic of exponential functions and their transformations, we recommend the following resources:

By understanding the transformation of the function $f(x)=2^x$ to obtain a new function $g(x)$ , you can gain insights into the behavior of exponential functions and their applications in various fields.