To solve and analyze the rational function $f(x) = \frac{2}{x} - 1$ step by step, we'll break it down into key aspects: domain, simplifying, finding asymptotes, intercepts, and the behavior of the function.

Step 1: Identify the Domain

The domain of the function is all the possible values of $x$ for which the function is defined. Since there is a fraction with $x$ in the denominator, we must ensure that $x$ is not equal to 0 (because division by zero is undefined).

Domain: $x \in \mathbb{R}$ except $x \neq 0$

Step 2: Simplify the Function (if possible)

The given function is $f(x) = \frac{2}{x} - 1$. This function is already simplified; no further simplification is necessary.

Step 3: Find Asymptotes

Vertical Asymptote: Vertical asymptotes occur where the denominator of the rational function equals zero, which makes the function undefined.

• Set the denominator equal to zero: $x = 0$
• So, there is a vertical asymptote at $x = 0$.

Horizontal Asymptote: For rational functions, the horizontal asymptote depends on the degrees of the polynomial in the numerator and denominator.

• The degree of the numerator is 0 (constant 2) and the degree of the denominator is 1 (since $x$ is in the denominator).
• Since the degree of the denominator is greater, the horizontal asymptote is $y = 0$.

Step 4: Find the Intercepts

x-intercept: The x-intercept occurs where $f(x) = 0$.

• Set the function equal to 0: $\frac{2}{x} - 1 = 0$
• Solve for $x$: $\frac{2}{x} = 1 \implies x = 2$
• So, the x-intercept is at $(2, 0)$.

y-intercept: The y-intercept occurs where $x = 0$, but since $x = 0$ is not in the domain, the function has no y-intercept.

Step 5: Analyze the Function's Behavior

To understand how the function behaves as $x$ approaches the asymptotes and other critical points:

• As $x$ approaches 0 from the right $(x \to 0^+)$, $f(x) \to +\infty$.
• As $x$ approaches 0 from the left $(x \to 0^-)$, $f(x) \to -\infty$.
• As $x$ approaches $\infty$, $f(x) \to -1$.
• As $x$ approaches $-\infty$, $f(x) \to -1$.

This means the function gets closer to -1 as $x$ increases or decreases in magnitude, but it never actually reaches -1.

Summary of the Function's Properties

• Domain: $x \in \mathbb{R}, x \neq 0$
• Vertical Asymptote: $x = 0$
• Horizontal Asymptote: $y = -1$
• x-intercept: $(2, 0)$
• y-intercept: None

Would you like to go over any specific part of this solution, or do you have any questions?

Relative Questions

1. How do you determine the horizontal asymptote of a rational function?
2. What happens to the value of $f(x)$ as $x$ approaches infinity?
3. How would you graph the function $f(x) = \frac{2}{x} - 1$?
4. Can a function have more than one vertical asymptote? Under what conditions?
5. What are the general steps to finding the intercepts of a rational function?

Tip

When analyzing rational functions, always check for both vertical and horizontal asymptotes to understand the function's behavior at extreme values of $x$.