Domain and Range of Functions

Oct 13, 2025 by wbrum.org 111 views

In mathematics, functions are used to describe the relationship between variables. Two functions, $f(x)$ and $g(x)$ , are given as $f(x)=\left(\frac{1}{2}\right)^x$ and $g(x)=\left(\frac{1}{2}\right)^{x+4}-2$ . In this article, we will analyze the properties of these functions and determine which statements about their functions are true.

Domain and Range of Functions

The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values.

Domain of $f(x)$

The function $f(x)=\left(\frac{1}{2}\right)^x$ is defined for all real numbers, as the base $\frac{1}{2}$ is positive and the exponent $x$ can take any real value. Therefore, the domain of $f(x)$ is all real numbers, denoted as $\mathbb{R}$ .

Domain of $g(x)$

The function $g(x)=\left(\frac{1}{2}\right)^{x+4}-2$ is also defined for all real numbers, as the base $\frac{1}{2}$ is positive and the exponent $x+4$ can take any real value. Therefore, the domain of $g(x)$ is also all real numbers, denoted as $\mathbb{R}$ .

Range of $f(x)$

The function $f(x)=\left(\frac{1}{2}\right)^x$ is an exponential function with a base of $\frac{1}{2}$ . As $x$ increases, the value of $f(x)$ decreases, and as $x$ decreases, the value of $f(x)$ increases. The range of $f(x)$ is all positive real numbers, denoted as $(0, \infty)$ .

Range of $g(x)$

The function $g(x)=\left(\frac{1}{2}\right)^{x+4}-2$ is also an exponential function with a base of $\frac{1}{2}$ . However, the range of $g(x)$ is shifted down by 2 units due to the constant term $-2$ . Therefore, the range of $g(x)$ is all real numbers less than 2, denoted as $(-\infty, 2)$ .

Comparing the Domains and Ranges of $f(x)$ and $g(x)$

From the analysis above, we can conclude that:

The domains of both functions are the same, which is all real numbers, denoted as $\mathbb{R}$ .
The ranges of both functions are not the same. The range of $f(x)$ is all positive real numbers, denoted as $(0, \infty)$ , while the range of $g(x)$ is all real numbers less than 2, denoted as $(-\infty, 2)$ .

Therefore, the correct answer is:

A. The domains of both functions are the same.
B. The ranges of both functions are not the same.

Conclusion

In conclusion, the analysis of the functions $f(x)=\left(\frac{1}{2}\right)^x$ and $g(x)=\left(\frac{1}{2}\right)^{x+4}-2$ has shown that the domains of both functions are the same, while the ranges of both functions are not the same. This understanding is crucial in mathematics, as it helps us to analyze and compare different functions and their properties.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Algebra, 2nd edition, Michael Artin

Additional Resources

Khan Academy: Functions
MIT OpenCourseWare: Calculus
Wolfram Alpha: Function Analysis
Q&A: Functions and Their Properties =====================================

In the previous article, we analyzed the properties of the functions $f(x)=\left(\frac{1}{2}\right)^x$ and $g(x)=\left(\frac{1}{2}\right)^{x+4}-2$ . In this article, we will answer some frequently asked questions about functions and their properties.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values.

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

A: To determine the domain and range of a function, you need to analyze the function's properties, such as its base, exponent, and any constant terms. For example, if a function has a base of 2 and an exponent of x, its domain is all real numbers, and its range is all positive 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, while the range of a function is the set of all possible output values. In other words, the domain is the input, and the range is the output.

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same. For example, if a function has a base of 2 and an exponent of x, its domain and range are both all positive real numbers.

Q: How do you compare the domains and ranges of two functions?

A: To compare the domains and ranges of two functions, you need to analyze their properties and determine if they are the same or different. For example, if two functions have the same base and exponent, but one has a constant term and the other does not, their domains and ranges may be different.

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

A: The domain and range of a function are significant because they help us to understand the function's behavior and properties. For example, if a function has a limited domain, it may not be defined for certain input values, and if a function has a limited range, it may not produce certain output values.

Q: Can the domain and range of a function be changed?

A: Yes, the domain and range of a function can be changed by modifying the function's properties, such as its base, exponent, and constant terms. For example, if a function has a base of 2 and an exponent of x, you can change its domain and range by changing the base or exponent.

Q: How do you graph a function?

A: To graph a function, you need to plot its input and output values on a coordinate plane. For example, if a function has a base of 2 and an exponent of x, you can graph it by plotting its input values (x) on the x-axis and its output values (y) on the y-axis.

Q: What is the significance of graphing a function?

A: Graphing a function is significant because it helps us to visualize its behavior and properties. For example, if a function has a limited domain or range, its graph may show a limited range of input or output values.

Conclusion

In conclusion, the domain and range of a function are important properties that help us to understand its behavior and properties. By analyzing the domain and range of a function, we can determine its input and output values, compare its properties with other functions, and graph its behavior.

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Algebra, 2nd edition, Michael Artin
[3] Khan Academy: Functions
[4] MIT OpenCourseWare: Calculus
[5] Wolfram Alpha: Function Analysis

Additional Resources

Khan Academy: Graphing Functions
MIT OpenCourseWare: Graphing Functions
Wolfram Alpha: Function Graphing