How Does The Range Of $g(x)=\frac{6}{x}$ Compare With The Range Of The Parent Function $f(x)=\frac{1}{x}$?A. The Range Of Both $f(x$\] And $g(x$\] Is All Real Numbers.B. The Range Of Both $f(x$\] And

Comparing the Ranges of $g(x)=\frac{6}{x}$ and $f(x)=\frac{1}{x}$

When dealing with functions, understanding the range of a function is crucial in determining its behavior and characteristics. In this article, we will explore the range of the function $g(x)=\frac{6}{x}$ and compare it with the range of its parent function $f(x)=\frac{1}{x}$. We will delve into the properties of these functions, analyze their behavior, and determine the range of each function.

Understanding the Parent Function $f(x)=\frac{1}{x}$

The parent function $f(x)=\frac{1}{x}$ is a reciprocal function, which means it has a reciprocal relationship with its input variable $x$. This function is defined for all real numbers except $x=0$, as division by zero is undefined.

Properties of $f(x)=\frac{1}{x}$

The function $f(x)=\frac{1}{x}$ has several key properties that are essential in understanding its behavior:

• Domain: The domain of $f(x)=\frac{1}{x}$ is all real numbers except $x=0$.
• Range: The range of $f(x)=\frac{1}{x}$ is all real numbers except $0$.
• Asymptotes: The function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.
• Symmetry: The function is symmetric about the origin.

Understanding the Function $g(x)=\frac{6}{x}$

The function $g(x)=\frac{6}{x}$ is a transformation of the parent function $f(x)=\frac{1}{x}$. It is also a reciprocal function, but with a different constant in the numerator.

Properties of $g(x)=\frac{6}{x}$

The function $g(x)=\frac{6}{x}$ has several key properties that are essential in understanding its behavior:

• Domain: The domain of $g(x)=\frac{6}{x}$ is all real numbers except $x=0$.
• Range: The range of $g(x)=\frac{6}{x}$ is all real numbers except $0$.
• Asymptotes: The function has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.
• Symmetry: The function is symmetric about the origin.

Comparing the Ranges of $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$

Now that we have analyzed the properties of both functions, let's compare their ranges.

The range of $f(x)=\frac{1}{x}$ is all real numbers except $0$. This means that the function can take on any value except $0$.

The range of $g(x)=\frac{6}{x}$ is also all real numbers except $0$. This means that the function can take on any value except $0$.

In conclusion, the range of both $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$ is all real numbers except $0$. This means that both functions can take on any value except $0$.

The final answer is that the range of both $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$ is all real numbers except $0$.

• [1] "Functions" by Khan Academy
• [2] "Reciprocal Functions" by Math Open Reference
• [3] "Domain and Range" by Purplemath

• [1] "Functions" by IXL
• [2] "Reciprocal Functions" by Mathway
• [3] "Domain and Range" by Math Open Reference
  Q&A: Comparing the Ranges of $g(x)=\frac{6}{x}$ and $f(x)=\frac{1}{x}$ ====================================================================

Here are some frequently asked questions about comparing the ranges of $g(x)=\frac{6}{x}$ and $f(x)=\frac{1}{x}$:

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

A: The range of $f(x)=\frac{1}{x}$ is all real numbers except $0$.

Q: What is the range of $g(x)=\frac{6}{x}$?

A: The range of $g(x)=\frac{6}{x}$ is all real numbers except $0$.

Q: How do the ranges of $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$ compare?

A: The ranges of $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$ are the same, both being all real numbers except $0$.

Q: Why is the range of $f(x)=\frac{1}{x}$ all real numbers except $0$?

A: The range of $f(x)=\frac{1}{x}$ is all real numbers except $0$ because the function can take on any value except $0$. This is due to the fact that the function is a reciprocal function, which means it has a reciprocal relationship with its input variable $x$.

Q: Why is the range of $g(x)=\frac{6}{x}$ all real numbers except $0$?

A: The range of $g(x)=\frac{6}{x}$ is all real numbers except $0$ because the function is a transformation of the parent function $f(x)=\frac{1}{x}$. This means that the function has the same properties as the parent function, including the same range.

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

A: The range of a function is significant because it determines the possible output values of the function. In other words, it determines the values that the function can take on.

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

A: To determine the range of a function, you can analyze the function's properties, such as its domain, asymptotes, and symmetry. You can also use mathematical techniques, such as graphing and algebraic manipulation, to determine the range of the function.

Q: What are some common mistakes to avoid when comparing the ranges of functions?

A: Some common mistakes to avoid when comparing the ranges of functions include:

• Assuming that the range of a function is all real numbers without analyzing the function's properties.
• Failing to consider the domain of a function when determining its range.
• Not accounting for asymptotes and symmetry when analyzing a function's range.

In conclusion, the range of both $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$ is all real numbers except $0$. This means that both functions can take on any value except $0$. By understanding the properties of these functions and analyzing their behavior, we can determine their ranges and make informed decisions about their applications.

The final answer is that the range of both $f(x)=\frac{1}{x}$ and $g(x)=\frac{6}{x}$ is all real numbers except $0$.

• [1] "Functions" by Khan Academy
• [2] "Reciprocal Functions" by Math Open Reference
• [3] "Domain and Range" by Purplemath

• [1] "Functions" by IXL
• [2] "Reciprocal Functions" by Mathway
• [3] "Domain and Range" by Math Open Reference