solve rational functions step by step which the given is f (x)=2/x-1

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 \( x \).

Discover how to solve and analyze the rational function f(x) = 2/(x - 1) step by step. Learn about its domain, vertical and horizontal asymptotes, intercepts, and behavior. Ideal for high school students studying advanced algebra.

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