The Functions $f(x$\] And $g(x$\] Are Defined Below.$\[ \begin{array}{r} f(x) = 4^{(x-2)} \\ g(x) = -2^x + 5 \end{array} \\]Using A Table Of Values, Determine The Solution To The Equation $f(x) = G(x$\].A. $x =

Feb 28, 2025 by ADMIN 211 views

**The Functions f(x) and g(x): A Table of Values Approach**

Introduction

In this article, we will explore the functions $f(x)$ and $g(x)$ , and use a table of values approach to determine the solution to the equation $f(x) = g(x)$ . The functions are defined as follows:

$f(x) = 4^{(x-2)}$

$g(x) = -2^x + 5$

Understanding the Functions

Before we proceed, let's take a closer look at the functions $f(x)$ and $g(x)$ .

Function f(x)

The function $f(x)$ is an exponential function with base 4. The exponent is $(x-2)$ , which means that the function will increase rapidly as $x$ increases. The function can be rewritten as:

$f(x) = 4^{(x-2)} = 4^x \cdot 4^{-2} = \frac{4^x}{16}$

Function g(x)

The function $g(x)$ is a linear function with a negative coefficient. The function can be rewritten as:

$g(x) = -2^x + 5$

Creating a Table of Values

To determine the solution to the equation $f(x) = g(x)$ , we will create a table of values for both functions. We will choose several values of $x$ and calculate the corresponding values of $f(x)$ and $g(x)$ .

$x$	$f(x)$	$g(x)$
-2	1	7
-1	2	5
0	4	5
1	8	3
2	16	1

Analyzing the Table of Values

From the table of values, we can see that the function $f(x)$ increases rapidly as $x$ increases, while the function $g(x)$ decreases as $x$ increases. We can also see that the two functions intersect at $x = 2$ .

Solution to the Equation

Based on the table of values, we can conclude that the solution to the equation $f(x) = g(x)$ is $x = 2$ .

Conclusion

In this article, we used a table of values approach to determine the solution to the equation $f(x) = g(x)$ . We created a table of values for both functions and analyzed the results to find the solution. The solution to the equation is $x = 2$ .

Additional Discussion

The functions $f(x)$ and $g(x)$ are both defined in terms of exponentials and linear functions. The function $f(x)$ is an exponential function with base 4, while the function $g(x)$ is a linear function with a negative coefficient. The equation $f(x) = g(x)$ can be solved using a table of values approach.

Final Thoughts

In conclusion, the functions $f(x)$ and $g(x)$ are both important concepts in mathematics. The function $f(x)$ is an exponential function with base 4, while the function $g(x)$ is a linear function with a negative coefficient. The equation $f(x) = g(x)$ can be solved using a table of values approach. The solution to the equation is $x = 2$ .

References

[1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-20.
[2] "Linear Functions". Math Open Reference. Retrieved 2023-02-20.

Appendix

The following is a list of additional resources that may be helpful in understanding the functions $f(x)$ and $g(x)$ .

[1] "Exponential Functions". Khan Academy. Retrieved 2023-02-20.
[2] "Linear Functions". Khan Academy. Retrieved 2023-02-20.

Introduction

In our previous article, we explored the functions $f(x)$ and $g(x)$ , and used a table of values approach to determine the solution to the equation $f(x) = g(x)$ . In this article, we will answer some frequently asked questions about the functions $f(x)$ and $g(x)$ .

Q&A

Q: What is the domain of the function f(x)?

A: The domain of the function $f(x)$ is all real numbers, since the base of the exponential function is 4, which is a positive number.

Q: What is the range of the function f(x)?

A: The range of the function $f(x)$ is all positive real numbers, since the base of the exponential function is 4, which is a positive number.

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

A: The domain of the function $g(x)$ is all real numbers, since the base of the exponential function is 2, which is a positive number.

Q: What is the range of the function g(x)?

A: The range of the function $g(x)$ is all real numbers, since the function is a linear function with a negative coefficient.

Q: How do I graph the functions f(x) and g(x)?

A: To graph the functions $f(x)$ and $g(x)$ , you can use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: How do I find the solution to the equation f(x) = g(x)?

A: To find the solution to the equation $f(x) = g(x)$ , you can use a table of values approach, as we did in our previous article.

Q: What is the significance of the solution to the equation f(x) = g(x)?

A: The solution to the equation $f(x) = g(x)$ is important because it shows the point at which the two functions intersect.

Q: Can I use other methods to solve the equation f(x) = g(x)?

A: Yes, you can use other methods to solve the equation $f(x) = g(x)$ , such as algebraic methods or numerical methods.

Conclusion

In this article, we answered some frequently asked questions about the functions $f(x)$ and $g(x)$ . We hope that this article has been helpful in understanding the functions and their properties.

Additional Discussion

The functions $f(x)$ and $g(x)$ are both important concepts in mathematics. The function $f(x)$ is an exponential function with base 4, while the function $g(x)$ is a linear function with a negative coefficient. The equation $f(x) = g(x)$ can be solved using a table of values approach.

Final Thoughts

References

[1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-20.
[2] "Linear Functions". Math Open Reference. Retrieved 2023-02-20.

Appendix

The following is a list of additional resources that may be helpful in understanding the functions $f(x)$ and $g(x)$ .

[1] "Exponential Functions". Khan Academy. Retrieved 2023-02-20.
[2] "Linear Functions". Khan Academy. Retrieved 2023-02-20.

Note: The references and appendix are not necessary for the solution to the equation, but they may be helpful in understanding the functions $f(x)$ and $g(x)$ in more detail.