Frequently Asked Questions about the Domain and Range of a Function

Oct 13, 2025 by windailys.com 92 views

Understanding the Domain and Range of a Function

When dealing with functions, it's essential to understand the concepts of domain and range. The domain of a function is the set of all possible input values for which the function is defined, while the range is the set of all possible output values. In this article, we'll explore the domain and range of the function $f(x)=\left(\frac{1}{10}\right)^x$ and determine which of the given statements is true.

What is the Domain of a Function?

The domain of a function is the set of all possible input values for which the function is defined. In other words, it's the set of all possible values of $x$ for which $f(x)$ is a real number. For the function $f(x)=\left(\frac{1}{10}\right)^x$ , we need to consider the values of $x$ that make the function defined.

Analyzing the Domain of $f(x)=\left(\frac{1}{10}\right)^x$

To determine the domain of $f(x)=\left(\frac{1}{10}\right)^x$ , we need to consider the values of $x$ that make the function defined. Since the function involves exponentiation, we need to ensure that the base $\frac{1}{10}$ is positive and the exponent $x$ is a real number.

The base $\frac{1}{10}$ is a positive number, so it's always defined. However, the exponent $x$ can be any real number, including negative numbers, zero, and positive numbers. Therefore, the domain of $f(x)=\left(\frac{1}{10}\right)^x$ is all real numbers, denoted as $(-\infty, \infty)$ .

What is the Range of a Function?

The range of a function is the set of all possible output values for which the function is defined. In other words, it's the set of all possible values of $f(x)$ for which $x$ is in the domain. For the function $f(x)=\left(\frac{1}{10}\right)^x$ , we need to consider the values of $f(x)$ that make the function defined.

Analyzing the Range of $f(x)=\left(\frac{1}{10}\right)^x$

To determine the range of $f(x)=\left(\frac{1}{10}\right)^x$ , we need to consider the values of $f(x)$ that make the function defined. Since the function involves exponentiation, we need to ensure that the base $\frac{1}{10}$ is positive and the exponent $x$ is a real number.

The base $\frac{1}{10}$ is a positive number, so it's always defined. However, the exponent $x$ can be any real number, including negative numbers, zero, and positive numbers. Therefore, the range of $f(x)=\left(\frac{1}{10}\right)^x$ is all positive real numbers, denoted as $(0, \infty)$ .

Evaluating the Statements

Now that we've determined the domain and range of the function $f(x)=\left(\frac{1}{10}\right)^x$ , let's evaluate the given statements.

A. The domain of $f(x)$ is $x > 0$ .

This statement is false because the domain of $f(x)$ is all real numbers, not just positive numbers.

B. The range of $f(x)$ is $y > \frac{1}{10}$ .

This statement is false because the range of $f(x)$ is all positive real numbers, not just numbers greater than $\frac{1}{10}$ .

C. The graph of $f(x)$ is a decreasing function.

This statement is true because the function $f(x)=\left(\frac{1}{10}\right)^x$ is an exponential function with a base less than 1, which means it's a decreasing function.

Conclusion

In conclusion, the domain of the function $f(x)=\left(\frac{1}{10}\right)^x$ is all real numbers, and the range is all positive real numbers. The correct statement is C, which states that the graph of $f(x)$ is a decreasing function.

Key Takeaways

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 for which the function is defined.
The function $f(x)=\left(\frac{1}{10}\right)^x$ is an exponential function with a base less than 1, which means it's a decreasing function.

Final Thoughts

Understanding the domain and range of a function is crucial in mathematics and other fields. By analyzing the domain and range of a function, we can determine its behavior and make predictions about its output values. In this article, we've explored the domain and range of the function $f(x)=\left(\frac{1}{10}\right)^x$ and determined which of the given statements is true.
Frequently Asked Questions about the Domain and Range of a Function

In the previous article, we explored the domain and range of the function $f(x)=\left(\frac{1}{10}\right)^x$ and determined which of the given statements is true. In this article, we'll answer some frequently asked questions about the domain and range of a function.

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 for which the function is defined, while the range is the set of all possible output values for which the function is defined.

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 the input variable that make the function defined. For example, if the function involves division, you need to ensure that the denominator is not equal to zero.

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

A: To determine the range of a function, you need to consider the values of the output variable that make the function defined. For example, if the function involves exponentiation, you need to ensure that the base is positive and the exponent is a real number.

Q: What is the domain of the function $f(x)=\frac{1}{x}$ ?

A: The domain of the function $f(x)=\frac{1}{x}$ is all real numbers except zero, denoted as $(-\infty, 0) \cup (0, \infty)$ .

Q: What is the range of the function $f(x)=\frac{1}{x}$ ?

A: The range of the function $f(x)=\frac{1}{x}$ is all positive real numbers, denoted as $(0, \infty)$ .

Q: What is the domain of the function $f(x)=x^2$ ?

A: The domain of the function $f(x)=x^2$ is all real numbers, denoted as $(-\infty, \infty)$ .

Q: What is the range of the function $f(x)=x^2$ ?

A: The range of the function $f(x)=x^2$ is all non-negative real numbers, denoted as $[0, \infty)$ .

Q: How do I determine if a function is increasing or decreasing?

A: To determine if a function is increasing or decreasing, you need to consider the behavior of the function as the input variable changes. If the function's output value increases as the input value increases, the function is increasing. If the function's output value decreases as the input value increases, the function is decreasing.

Q: What is the difference between an increasing and decreasing function?

A: An increasing function is a function whose output value increases as the input value increases, while a decreasing function is a function whose output value decreases as the input value increases.

Q: Can a function be both increasing and decreasing?

A: No, a function cannot be both increasing and decreasing. A function is either increasing or decreasing, but not both.

Q: How do I graph a function?

A: To graph a function, you need to plot the points on the coordinate plane that satisfy the function's equation. You can use a graphing calculator or software to help you graph the function.

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

A: The domain and range of a function are important because they determine the function's behavior and output values. Understanding the domain and range of a function is crucial in mathematics and other fields.

Conclusion

In conclusion, the domain and range of a function are crucial concepts in mathematics and other fields. By understanding the domain and range of a function, you can determine its behavior and make predictions about its output values. In this article, we've answered some frequently asked questions about the domain and range of a function.