Consider the rational expression f(x)=5x−1−x+2f(x)=5x−1−x+2, What is its horizontal asymptote? y =

The given rational expression appears to be:  

\[
f(x) = \frac{5x - 1}{-x + 2}
\]

### Step 1: Identify the Degrees of the Polynomials  
A rational function is of the form:

\[
f(x) = \frac{P(x)}{Q(x)}
\]

where \( P(x) \) is the numerator and \( Q(x) \) is the denominator.  
- The degree of the numerator \( P(x) = 5x - 1 \) is **1**.  
- The degree of the denominator \( Q(x) = -x + 2 \) is **1**.  

### Step 2: Determine the Horizontal Asymptote  
The horizontal asymptote for a rational function follows these rules:  
1. If the degree of the numerator is **less than** the degree of the denominator, the horizontal asymptote is \( y = 0 \).  
2. If the degree of the numerator is **equal to** the degree of the denominator, the horizontal asymptote is:

   \[
   y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}
   \]

   Here, the leading coefficient of \( 5x - 1 \) is **5**, and the leading coefficient of \( -x + 2 \) is **-1**.  

   \[
   y = \frac{5}{-1} = -5
   \]

### Final Answer:  
The horizontal asymptote is:

\[
y = -5
\]

---

Would you like further explanation or examples?  

### Related Questions:  
1. How do you find the vertical asymptotes of \( f(x) = \frac{5x - 1}{-x + 2} \)?  
2. What happens to the function as \( x \to \infty \) and \( x \to -\infty \)?  
3. How do you determine if a rational function has a slant asymptote?  
4. Can a rational function have more than one horizontal asymptote?  
5. How do you find the x- and y-intercepts of \( f(x) \)?  

### Tip:  
If the numerator's degree is exactly **one greater** than the denominator's, there is no horizontal asymptote, but a **slant asymptote** found using polynomial division.

Consider the rational expression f(x) = (5x - 1) / (-x + 2). What is its horizontal asymptote? y =

Learn how to determine the horizontal asymptote of the rational function f(x) = (5x - 1) / (-x + 2). This step-by-step solution explains the concept of asymptotes, key formulas, and theorems. Suitable for students in Grades 9-12 studying algebra and calculus.

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