Determine The Domain Of The Function: $\[ G(x) = 2^{(x-1)} \\]Complete The Table:$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline -3 & \\ \hline -2 & \\ \hline -1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 &

Feb 26, 2025 by ADMIN 226 views

**Determining the Domain of a Function: A Comprehensive Guide**

Introduction

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. In this article, we'll focus on determining the domain of the function $g(x) = 2^{(x-1)}$ .

Understanding the Function

The given function is $g(x) = 2^{(x-1)}$ . This is an exponential function with base 2, where the exponent is $(x-1)$ . To determine the domain of this function, we need to consider the values of x that make the function defined.

The Concept of Domain

The domain of a function is a set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. The domain of a function can be represented using interval notation, which includes all the possible x-values.

Determining the Domain of the Function

To determine the domain of the function $g(x) = 2^{(x-1)}$ , we need to consider the values of x that make the function defined. Since the function is an exponential function with base 2, the exponent $(x-1)$ must be a real number. This means that the function is defined for all real numbers x.

Completing the Table

To complete the table, we need to find the values of y for each value of x. We can do this by plugging in the values of x into the function $g(x) = 2^{(x-1)}$ .

x	y
-3	$2^{(-3-1)} = 2^{-4} = \frac{1}{16}$
-2	$2^{(-2-1)} = 2^{-3} = \frac{1}{8}$
-1	$2^{(-1-1)} = 2^{-2} = \frac{1}{4}$
0	$2^{(0-1)} = 2^{-1} = \frac{1}{2}$
1	$2^{(1-1)} = 2^{0} = 1$
2	$2^{(2-1)} = 2^{1} = 2$
3	$2^{(3-1)} = 2^{2} = 4$

Analyzing the Results

From the completed table, we can see that the function $g(x) = 2^{(x-1)}$ is defined for all real numbers x. This means that the domain of the function is the set of all real numbers.

Conclusion

In conclusion, the domain of the function $g(x) = 2^{(x-1)}$ is the set of all real numbers. This means that the function is defined for all possible input values (x-values) without resulting in an undefined or imaginary output. The completed table provides a visual representation of the function and its domain.

Discussion

The concept of domain is crucial in mathematics, as it helps us understand the behavior of functions and their limitations. In this article, we've determined the domain of the function $g(x) = 2^{(x-1)}$ and completed the table to provide a visual representation of the function and its domain.

Real-World Applications

The concept of domain has numerous real-world applications, including:

Computer Science: In computer science, the domain of a function is used to determine the input values that a function can accept without resulting in an error or undefined output.
Engineering: In engineering, the domain of a function is used to determine the input values that a function can accept without resulting in an unstable or undefined output.
Economics: In economics, the domain of a function is used to determine the input values that a function can accept without resulting in an undefined or imaginary output.

Final Thoughts

In conclusion, the domain of a function is a crucial concept in mathematics that helps us understand the behavior of functions and their limitations. In this article, we've determined the domain of the function $g(x) = 2^{(x-1)}$ and completed the table to provide a visual representation of the function and its domain. The concept of domain has numerous real-world applications, including computer science, engineering, and economics.

References

Khan Academy: Khan Academy provides a comprehensive guide to functions and their domains.
Math Is Fun: Math Is Fun provides a detailed explanation of the concept of domain and its applications.
Wolfram MathWorld: Wolfram MathWorld provides a comprehensive guide to functions and their domains, including the concept of domain and its applications.

Introduction

Determining the domain of a function is a crucial concept in mathematics that helps us understand the behavior of functions and their limitations. In our previous article, we determined the domain of the function $g(x) = 2^{(x-1)}$ and completed the table to provide a visual representation of the function and its domain. In this article, we'll answer some frequently asked questions about determining the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the values of x that make the function defined. This involves analyzing the function and identifying any restrictions or limitations on the input values.

Q: What are some common restrictions on the domain of a function?

A: Some common restrictions on the domain of a function include:

Division by zero: If a function involves division by zero, the domain is restricted to all real numbers except the value that would result in division by zero.
Square root: If a function involves a square root, the domain is restricted to all real numbers except negative values.
Exponents: If a function involves exponents, the domain is restricted to all real numbers except values that would result in an undefined or imaginary output.

Q: How do I represent the domain of a function?

A: The domain of a function can be represented using interval notation, which includes all the possible x-values. For example, the domain of the function $g(x) = 2^{(x-1)}$ is represented as $(-\infty, \infty)$ , indicating that the function is defined for all real numbers.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

Q: How do I find the range of a function?

A: To find the range of a function, you need to analyze the function and identify the possible output values. This involves considering the domain of the function and the behavior of the function as the input values change.

Q: What are some real-world applications of determining the domain of a function?

A: Determining the domain of a function has numerous real-world applications, including:

Computer Science: In computer science, the domain of a function is used to determine the input values that a function can accept without resulting in an error or undefined output.
Engineering: In engineering, the domain of a function is used to determine the input values that a function can accept without resulting in an unstable or undefined output.
Economics: In economics, the domain of a function is used to determine the input values that a function can accept without resulting in an undefined or imaginary output.

Q: How do I determine the domain of a function with multiple variables?

A: To determine the domain of a function with multiple variables, you need to consider the values of each variable that make the function defined. This involves analyzing the function and identifying any restrictions or limitations on the input values.

Conclusion

Determining the domain of a function is a crucial concept in mathematics that helps us understand the behavior of functions and their limitations. By understanding the domain of a function, you'll be able to analyze and solve problems involving functions and their limitations. We hope this article has provided you with a comprehensive guide to determining the domain of a function.

References

Khan Academy: Khan Academy provides a comprehensive guide to functions and their domains.
Math Is Fun: Math Is Fun provides a detailed explanation of the concept of domain and its applications.
Wolfram MathWorld: Wolfram MathWorld provides a comprehensive guide to functions and their domains, including the concept of domain and its applications.

Determine The Domain Of The Function: $\[ G(x) = 2^{(x-1)} \\]Complete The Table:$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline -3 & \\ \hline -2 & \\ \hline -1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 &

Introduction

Understanding the Function

The Concept of Domain

Determining the Domain of the Function

Completing the Table

Analyzing the Results

Conclusion

Discussion

Real-World Applications

Final Thoughts

References

Further Reading

Introduction

Q: What is the domain of a function?

Q: How do I determine the domain of a function?

Q: What are some common restrictions on the domain of a function?

Q: How do I represent the domain of a function?

Q: What is the difference between the domain and range of a function?

Q: How do I find the range of a function?

Q: What are some real-world applications of determining the domain of a function?

Q: How do I determine the domain of a function with multiple variables?

Conclusion

References

Further Reading