Domain and Range Comparison

Oct 13, 2025 by miodeya.com 55 views

When dealing with functions, it's essential to understand their domain and range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we'll explore the domain and range of three given functions: $f(x)=3x^2$ , $g(x)=\frac{1}{3x}$ , and $h(x)=3x$ .

Understanding the Domain and Range

Before we dive into the specific functions, let's recall the definitions of domain and range.

Domain: The set of all possible input values for a function. In other words, it's the set of all values of x that can be plugged into the function.
Range: The set of all possible output values for a function. In other words, it's the set of all values of y that can be obtained by plugging in different values of x.

Function 1: $f(x)=3x^2$

The function $f(x)=3x^2$ is a quadratic function. To find its domain and range, we need to consider the following:

Domain: Since the function involves squaring x, it's defined for all real numbers. Therefore, the domain of $f(x)$ is all real numbers, denoted as $(-\infty, \infty)$ .
Range: The function $f(x)=3x^2$ is always positive, since the square of any real number is non-negative, and multiplying it by 3 makes it positive. Therefore, the range of $f(x)$ is all positive real numbers, denoted as $(0, \infty)$ .

Function 2: $g(x)=\frac{1}{3x}$

The function $g(x)=\frac{1}{3x}$ is a rational function. To find its domain and range, we need to consider the following:

Domain: Since the function involves division by x, it's undefined when x is equal to 0. Therefore, the domain of $g(x)$ is all real numbers except 0, denoted as $(-\infty, 0) \cup (0, \infty)$ .
Range: The function $g(x)=\frac{1}{3x}$ can take on any positive value, since the reciprocal of any non-zero real number is positive. Therefore, the range of $g(x)$ is all positive real numbers, denoted as $(0, \infty)$ .

Function 3: $h(x)=3x$

The function $h(x)=3x$ is a linear function. To find its domain and range, we need to consider the following:

Domain: Since the function involves multiplication by 3, it's defined for all real numbers. Therefore, the domain of $h(x)$ is all real numbers, denoted as $(-\infty, \infty)$ .
Range: The function $h(x)=3x$ can take on any real value, since multiplying any real number by 3 results in another real number. Therefore, the range of $h(x)$ is all real numbers, denoted as $(-\infty, \infty)$ .

Comparing the Domain and Range

Now that we've analyzed each function, let's compare their domain and range.

Unique Range: Only the function $f(x)=3x^2$ has a unique range, which is all positive real numbers. The other two functions have a range of all real numbers.
Domain and Range Comparison: The function $g(x)=\frac{1}{3x}$ has a domain of all real numbers except 0, while the other two functions have a domain of all real numbers. The function $h(x)=3x$ has a range of all real numbers, while the other two functions have a range of either all positive real numbers or all real numbers.

Conclusion

In conclusion, the domain and range of the three functions $f(x)=3x^2$ , $g(x)=\frac{1}{3x}$ , and $h(x)=3x$ are as follows:

Function $f(x)=3x^2$ : Domain = $(-\infty, \infty)$ , Range = $(0, \infty)$
Function $g(x)=\frac{1}{3x}$ : Domain = $(-\infty, 0) \cup (0, \infty)$ , Range = $(0, \infty)$
Function $h(x)=3x$ : Domain = $(-\infty, \infty)$ , Range = $(-\infty, \infty)$

In the previous article, we explored the domain and range of three functions: $f(x)=3x^2$ , $g(x)=\frac{1}{3x}$ , and $h(x)=3x$ . Now, let's answer some frequently asked questions about the domain and range of functions.

Q1: What is the domain of a function?

A1: The domain of a function is the set of all possible input values for the function. In other words, it's the set of all values of x that can be plugged into the function.

Q2: What is the range of a function?

A2: The range of a function is the set of all possible output values for the function. In other words, it's the set of all values of y that can be obtained by plugging in different values of x.

Q3: How do I determine the domain of a function?

A3: To determine the domain of a function, you need to consider the following:

Squaring or taking the square root: If the function involves squaring or taking the square root of x, the domain is all real numbers.
Division: If the function involves division by x, the domain is all real numbers except 0.
Multiplication or addition: If the function involves multiplication or addition, the domain is all real numbers.

Q4: How do I determine the range of a function?

A4: To determine the range of a function, you need to consider the following:

Squaring or taking the square root: If the function involves squaring or taking the square root of x, the range is all non-negative real numbers.
Division: If the function involves division by x, the range is all positive real numbers.
Multiplication or addition: If the function involves multiplication or addition, the range is all real numbers.

Q5: Can a function have a unique range?

A5: Yes, a function can have a unique range. For example, the function $f(x)=3x^2$ has a unique range of all positive real numbers.

Q6: Can a function have a domain of all real numbers?

A6: Yes, a function can have a domain of all real numbers. For example, the functions $f(x)=3x^2$ and $h(x)=3x$ have a domain of all real numbers.

Q7: Can a function have a range of all real numbers?

A7: Yes, a function can have a range of all real numbers. For example, the function $h(x)=3x$ has a range of all real numbers.

Q8: How do I compare the domain and range of two functions?

A8: To compare the domain and range of two functions, you need to consider the following:

Domain: Compare the domain of each function to determine which one is larger or smaller.
Range: Compare the range of each function to determine which one is larger or smaller.

Q9: Can a function have a domain and range of all real numbers?

A9: Yes, a function can have a domain and range of all real numbers. For example, the function $h(x)=3x$ has a domain and range of all real numbers.

Q10: Can a function have a unique domain and range?

A10: No, a function cannot have a unique domain and range. The domain and range of a function are always related to each other.

Conclusion

In conclusion, the domain and range of functions are essential concepts in mathematics. By understanding the domain and range of a function, you can better analyze and compare its properties. We hope this Q&A article has helped you understand the domain and range of functions better.