E) Sketch F ( X ) = X 2 − 9 X − 2 F(x)=\frac{x^2-9}{x-2} F ( X ) = X − 2 X 2 − 9 And Determine Its Range And Domain.

Mar 9, 2025 by ADMIN 119 views

e) Sketch $f(x)=\frac{x^2-9}{x-2}$ and determine its range and domain

In this article, we will explore the process of sketching a rational function and determining its domain and range. The given function is $f(x)=\frac{x^2-9}{x-2}$ . We will start by factoring the numerator and simplifying the function. Then, we will analyze the behavior of the function as $x$ approaches positive and negative infinity, and identify any vertical asymptotes. Finally, we will determine the domain and range of the function.

Factoring the Numerator

The numerator of the function can be factored as follows:

$x^2 - 9 = (x + 3)(x - 3)$

So, the function can be rewritten as:

$f(x) = \frac{(x + 3)(x - 3)}{x - 2}$

Simplifying the Function

We can simplify the function by canceling out the common factor of $(x - 2)$ from the numerator and denominator:

$f(x) = x + 3$

However, we must note that this simplification is only valid when $x \neq 2$ , since the original function is undefined at $x = 2$ .

Analyzing the Behavior of the Function

As $x$ approaches positive infinity, the function $f(x) = x + 3$ also approaches positive infinity. Similarly, as $x$ approaches negative infinity, the function approaches negative infinity.

Identifying Vertical Asymptotes

The function has a vertical asymptote at $x = 2$ , since the denominator is equal to zero at this point. This means that the function is undefined at $x = 2$ .

Determining the Domain

The domain of the function is all real numbers except $x = 2$ . This is because the function is undefined at $x = 2$ , due to the vertical asymptote.

Determining the Range

The range of the function is all real numbers except $y = 5$ . This is because the function approaches positive infinity as $x$ approaches positive infinity, and approaches negative infinity as $x$ approaches negative infinity. However, the function is never equal to $y = 5$ , since this would require $x = 2$ , which is not in the domain of the function.

Sketching the Function

The graph of the function is a straight line with a slope of 1 and a y-intercept of 3. However, the graph has a hole at $x = 2$ , since the function is undefined at this point.

In conclusion, we have sketched the function $f(x)=\frac{x^2-9}{x-2}$ and determined its domain and range. The function has a vertical asymptote at $x = 2$ and a hole at this point. The domain of the function is all real numbers except $x = 2$ , and the range is all real numbers except $y = 5$ .

Key Takeaways

The function $f(x)=\frac{x^2-9}{x-2}$ can be simplified to $f(x) = x + 3$ when $x \neq 2$ .
The function has a vertical asymptote at $x = 2$ and a hole at this point.
The domain of the function is all real numbers except $x = 2$ .
The range of the function is all real numbers except $y = 5$ .

Final Thoughts

Sketching rational functions and determining their domain and range is an important skill in mathematics. By following the steps outlined in this article, you can sketch a rational function and determine its domain and range. Remember to identify any vertical asymptotes and holes in the graph, and to determine the domain and range of the function.
Q&A: Sketching $f(x)=\frac{x^2-9}{x-2}$ and Determining its Domain and Range

In our previous article, we explored the process of sketching a rational function and determining its domain and range. The given function was $f(x)=\frac{x^2-9}{x-2}$ . We simplified the function, analyzed its behavior as $x$ approaches positive and negative infinity, and identified any vertical asymptotes. We also determined the domain and range of the function.

In this article, we will answer some common questions related to sketching rational functions and determining their domain and range.

Q: What is the difference between a vertical asymptote and a hole in a graph?

A: A vertical asymptote is a vertical line that the graph approaches but never touches. A hole in a graph, on the other hand, is a point where the graph is undefined but approaches a specific value.

Q: How do I identify vertical asymptotes in a rational function?

A: To identify vertical asymptotes in a rational function, you need to find the values of $x$ that make the denominator equal to zero. These values are the vertical asymptotes of the function.

Q: What is the domain of a rational function?

A: The domain of a rational function is all real numbers except the values of $x$ that make the denominator equal to zero. These values are the vertical asymptotes of the function.

Q: What is the range of a rational function?

A: The range of a rational function is all real numbers except the values of $y$ that the function approaches but never touches. These values are the vertical asymptotes of the function.

Q: How do I sketch a rational function?

A: To sketch a rational function, you need to follow these steps:

Simplify the function by canceling out any common factors in the numerator and denominator.
Identify any vertical asymptotes by finding the values of $x$ that make the denominator equal to zero.
Analyze the behavior of the function as $x$ approaches positive and negative infinity.
Plot the graph of the function, including any vertical asymptotes and holes.

Q: What is the difference between a rational function and a polynomial function?

A: A rational function is a function that can be written in the form $f(x) = \frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials. A polynomial function, on the other hand, is a function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ , where $a_n \neq 0$ .

Q: Can a rational function have a hole in its graph?

A: Yes, a rational function can have a hole in its graph. A hole occurs when the function is undefined at a specific point, but approaches a specific value as $x$ approaches that point.

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

A: To determine the domain and range of a rational function, you need to follow these steps:

Simplify the function by canceling out any common factors in the numerator and denominator.
Identify any vertical asymptotes by finding the values of $x$ that make the denominator equal to zero.
Analyze the behavior of the function as $x$ approaches positive and negative infinity.
Determine the domain and range of the function based on the vertical asymptotes and holes in the graph.

In conclusion, we have answered some common questions related to sketching rational functions and determining their domain and range. We hope that this article has provided you with a better understanding of these concepts and how to apply them to real-world problems.