The problem presents the graph of the function $f : \mathbb{R} \to \mathbb{R}, f(x) = \frac{2x^3 - x - 1}{x^2 + 2}$, and asks to identify the correct statement regarding the asymptotes.

Let's break this down:

Step 1: Analyze the Function $f(x) = \frac{2x^3 - x - 1}{x^2 + 2}$

This is a rational function where:

• The degree of the numerator (highest power of $x$) is 3.
• The degree of the denominator is 2.

Since the degree of the numerator is greater than the degree of the denominator by 1, this means the function has an oblique (slant) asymptote.

Step 2: Find the Oblique Asymptote

To determine the equation of the oblique asymptote, divide the numerator by the denominator using polynomial long division:

$\frac{2x^3 - x - 1}{x^2 + 2} = 2x - \frac{5x + 1}{x^2 + 2}$

The quotient is $y = 2x$, which represents the oblique asymptote of the function.

Step 3: Verify the Choices

• First option: $y = 2x$ is an oblique asymptote — This is correct based on the division result.
• Second option: $y = -x + 3$ — This is incorrect.
• Third option: $y = 2x - 5$ — This is incorrect; although $y = 2x$ is correct, the vertical shift is wrong.
• Fourth option: $y = 2$ is a horizontal asymptote — This is incorrect because the function does not have a horizontal asymptote, only an oblique one.

Conclusion:

The correct answer is the first option: $y = 2x$ is the oblique asymptote.

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:

1. What is the procedure for finding vertical asymptotes in a rational function?
2. How can polynomial long division help in analyzing the behavior of rational functions at infinity?
3. What is the difference between horizontal and oblique asymptotes in terms of function degree?
4. How would you find the exact coordinates of the intersection between the function and its asymptote?
5. Can you explain the significance of oblique asymptotes in practical applications?

Tip: When solving rational functions, always check if the numerator's degree exceeds the denominator's to detect potential slant or oblique asymptotes!