Translation of the Basic Rational Function

Oct 13, 2025 by thesweetdesigns.com 50 views

Understanding the Function g(x) = 1/(x+3) - 5

In mathematics, functions are used to describe the relationship between variables. A function can be represented graphically, and understanding its behavior is crucial in various mathematical and real-world applications. The given function $g(x) = \frac{1}{x+3} - 5$ is a rational function, which is a type of function that has a variable in the denominator. In this article, we will analyze the given function and determine which of the provided statements is true.

The function $g(x) = \frac{1}{x+3} - 5$ can be broken down into two parts: the rational function $\frac{1}{x+3}$ and the constant function $-5$ . The rational function $\frac{1}{x+3}$ has a vertical asymptote at $x = -3$ , which means that the function approaches positive or negative infinity as $x$ approaches $-3$ from the left or right, respectively. The constant function $-5$ is a horizontal line that is 5 units below the x-axis.

Translation of the Basic Rational Function

The basic rational function $f(x) = \frac{1}{x}$ has a vertical asymptote at $x = 0$ . To translate this function, we need to consider the effects of the translation on the graph. If we translate the function $f(x) = \frac{1}{x}$ right by 3 units, the new function becomes $f(x-3) = \frac{1}{x-3}$ . This translation shifts the graph of the function to the right by 3 units.

Downward Translation

If we translate the function $f(x) = \frac{1}{x}$ downward by 5 units, the new function becomes $f(x) - 5 = \frac{1}{x} - 5$ . This translation shifts the graph of the function downward by 5 units.

Comparing the Original Function with the Translated Function

Now, let's compare the original function $g(x) = \frac{1}{x+3} - 5$ with the translated function $f(x-3) - 5 = \frac{1}{x-3} - 5$ . We can see that the original function has the same form as the translated function, but with a different denominator. The denominator of the original function is $x+3$ , while the denominator of the translated function is $x-3$ .

Based on the analysis, we can conclude that the original function $g(x) = \frac{1}{x+3} - 5$ can be graphed by translating the basic rational function $f(x) = \frac{1}{x}$ right by 3 units and downward by 5 units. This is because the original function has the same form as the translated function, but with a different denominator.

The correct answer is A) $g(x)$ can be graphed by translating the basic rational function $f(x) = \frac{1}{x}$ right by 3 units and downward by 5 units.

In conclusion, understanding the behavior of functions is crucial in mathematics and real-world applications. By analyzing the given function $g(x) = \frac{1}{x+3} - 5$ , we can determine which of the provided statements is true. The original function can be graphed by translating the basic rational function $f(x) = \frac{1}{x}$ right by 3 units and downward by 5 units.
Q&A: Understanding the Function g(x) = 1/(x+3) - 5

In our previous article, we analyzed the function $g(x) = \frac{1}{x+3} - 5$ and determined that it can be graphed by translating the basic rational function $f(x) = \frac{1}{x}$ right by 3 units and downward by 5 units. In this article, we will answer some frequently asked questions about the function $g(x) = \frac{1}{x+3} - 5$ .

Q: What is the domain of the function g(x) = 1/(x+3) - 5?

A: The domain of the function $g(x) = \frac{1}{x+3} - 5$ is all real numbers except $x = -3$ . This is because the function has a vertical asymptote at $x = -3$ , which means that the function approaches positive or negative infinity as $x$ approaches $-3$ from the left or right, respectively.

Q: What is the range of the function g(x) = 1/(x+3) - 5?

A: The range of the function $g(x) = \frac{1}{x+3} - 5$ is all real numbers except $-5$ . This is because the function has a horizontal asymptote at $y = -5$ , which means that the function approaches $-5$ as $x$ approaches positive or negative infinity.

Q: How do I graph the function g(x) = 1/(x+3) - 5?

A: To graph the function $g(x) = \frac{1}{x+3} - 5$ , you can start by graphing the basic rational function $f(x) = \frac{1}{x}$ . Then, translate the graph of $f(x)$ right by 3 units and downward by 5 units. This will give you the graph of the function $g(x) = \frac{1}{x+3} - 5$ .

Q: What is the equation of the vertical asymptote of the function g(x) = 1/(x+3) - 5?

A: The equation of the vertical asymptote of the function $g(x) = \frac{1}{x+3} - 5$ is $x = -3$ . This is because the function has a vertical asymptote at $x = -3$ , which means that the function approaches positive or negative infinity as $x$ approaches $-3$ from the left or right, respectively.

Q: What is the equation of the horizontal asymptote of the function g(x) = 1/(x+3) - 5?

A: The equation of the horizontal asymptote of the function $g(x) = \frac{1}{x+3} - 5$ is $y = -5$ . This is because the function has a horizontal asymptote at $y = -5$ , which means that the function approaches $-5$ as $x$ approaches positive or negative infinity.

Q: Can I simplify the function g(x) = 1/(x+3) - 5?

A: Yes, you can simplify the function $g(x) = \frac{1}{x+3} - 5$ by combining the two terms. This will give you the simplified function $g(x) = \frac{1}{x+3} - \frac{5(x+3)}{x+3} = \frac{-5x-15}{x+3}$ .

In conclusion, the function $g(x) = \frac{1}{x+3} - 5$ is a rational function that has a vertical asymptote at $x = -3$ and a horizontal asymptote at $y = -5$ . The function can be graphed by translating the basic rational function $f(x) = \frac{1}{x}$ right by 3 units and downward by 5 units. We hope that this Q&A article has helped you to understand the function $g(x) = \frac{1}{x+3} - 5$ better.