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 Range of $g(x)=\frac{6}{x}$ and the Parent Function $f(x)=\frac{1}{x}$

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 that it has a reciprocal relationship with its input variable $x$. This function has a characteristic "S" shape, with a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$. The range of the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero, as the function approaches infinity as $x$ approaches zero from either side.

Properties of the Parent Function

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

• Domain: The domain of the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero, as the function is undefined at $x=0$.
• Range: The range of the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero, as the function approaches infinity as $x$ approaches zero from either side.
• Vertical Asymptote: The parent function $f(x)=\frac{1}{x}$ has a vertical asymptote at $x=0$, which means that the function approaches infinity as $x$ approaches zero from either side.
• Horizontal Asymptote: The parent function $f(x)=\frac{1}{x}$ has a horizontal asymptote at $y=0$, which means that the function approaches zero as $x$ approaches infinity.

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

The function $g(x)=\frac{6}{x}$ is a transformation of the parent function $f(x)=\frac{1}{x}$. This function has a similar "S" shape to the parent function, but with a different vertical asymptote and horizontal asymptote. The range of the function $g(x)=\frac{6}{x}$ is also all real numbers except zero, as the function approaches infinity as $x$ approaches zero from either side.

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

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

• Domain: The domain of the function $g(x)=\frac{6}{x}$ is all real numbers except zero, as the function is undefined at $x=0$.
• Range: The range of the function $g(x)=\frac{6}{x}$ is all real numbers except zero, as the function approaches infinity as $x$ approaches zero from either side.
• Vertical Asymptote: The function $g(x)=\frac{6}{x}$ has a vertical asymptote at $x=0$, which means that the function approaches infinity as $x$ approaches zero from either side.
• Horizontal Asymptote: The function $g(x)=\frac{6}{x}$ has a horizontal asymptote at $y=0$, which means that the function approaches zero as $x$ approaches infinity.

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

The range of both $g(x)=\frac{6}{x}$ and the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero. This is because both functions approach infinity as $x$ approaches zero from either side, and both functions have a horizontal asymptote at $y=0$. Therefore, the range of both functions is the same.

In conclusion, the range of both $g(x)=\frac{6}{x}$ and the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero. This is because both functions approach infinity as $x$ approaches zero from either side, and both functions have a horizontal asymptote at $y=0$. Understanding the range of a function is crucial in determining its behavior and characteristics, and this article has provided a comprehensive analysis of the range of both functions.

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Function Range and Domain
  Q&A: Understanding the Range of $g(x)=\frac{6}{x}$ and the Parent Function $f(x)=\frac{1}{x}$

We have received several questions from readers regarding the range of $g(x)=\frac{6}{x}$ and the parent function $f(x)=\frac{1}{x}$. Below are some of the most frequently asked questions and their answers.

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

A: The range of the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero. This is because the function approaches infinity as $x$ approaches zero from either side, and the function has a horizontal asymptote at $y=0$.

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

A: The range of the function $g(x)=\frac{6}{x}$ is all real numbers except zero. This is because the function approaches infinity as $x$ approaches zero from either side, and the function has a horizontal asymptote at $y=0$.

Q: 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 $g(x)=\frac{6}{x}$ and the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero. This is because both functions approach infinity as $x$ approaches zero from either side, and both functions have a horizontal asymptote at $y=0$.

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

A: The domain of the parent function $f(x)=\frac{1}{x}$ is all real numbers except zero. This is because the function is undefined at $x=0$.

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

A: The domain of the function $g(x)=\frac{6}{x}$ is all real numbers except zero. This is because the function is undefined at $x=0$.

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

A: The vertical asymptote of the parent function $f(x)=\frac{1}{x}$ is $x=0$. This is because the function approaches infinity as $x$ approaches zero from either side.

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

A: The vertical asymptote of the function $g(x)=\frac{6}{x}$ is $x=0$. This is because the function approaches infinity as $x$ approaches zero from either side.

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

A: The horizontal asymptote of the parent function $f(x)=\frac{1}{x}$ is $y=0$. This is because the function approaches zero as $x$ approaches infinity.

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

A: The horizontal asymptote of the function $g(x)=\frac{6}{x}$ is $y=0$. This is because the function approaches zero as $x$ approaches infinity.

We hope that this Q&A article has provided you with a better understanding of the range of $g(x)=\frac{6}{x}$ and the parent function $f(x)=\frac{1}{x}$. If you have any further questions, please don't hesitate to contact us.

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus
• Wolfram Alpha: Function Range and Domain