AP Precalculus Practice Test: Unit 1 Question #20 Domain and Horizontal Asymptote of a Rational

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

In AP Precalculus, Unit 1 includes understanding the domain and horizontal asymptotes of rational functions. Question #20 likely focuses on analyzing these aspects to understand the function's behavior and restrictions.

Here’s the approach for determining the domain and horizontal asymptote of a rational function:

### 1. **Finding the Domain**
- A rational function is expressed as \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials.
- The domain includes all real numbers except where \( q(x) = 0 \), as dividing by zero is undefined.
- To find the domain, set the denominator \( q(x) \) equal to zero and solve for \( x \). The solutions are the values excluded from the domain.

### 2. **Finding the Horizontal Asymptote**
- The horizontal asymptote describes the behavior of the function as \( x \) approaches infinity.
- For rational functions, determine the horizontal asymptote by comparing the degrees (highest power of \( x \)) in the numerator and denominator:
- **If the degree of the numerator is less than the degree of the denominator**, the horizontal asymptote is \( y = 0 \).
- **If the degree of the numerator equals the degree of the denominator**, the horizontal asymptote is \( y = \frac{\text{leading coefficient of } p(x)}{\text{leading coefficient of } q(x)} \).
- **If the degree of the numerator is greater than the degree of the denominator**, there is no horizontal asymptote (though there may be an oblique asymptote).

---

### Example:
For a rational function \( f(x) = \frac{2x^2 + 3x - 5}{x^2 - 4} \):

1. **Domain**:
- Set the denominator equal to zero: \( x^2 - 4 = 0 \).
- Solving this, we get \( x = 2 \) and \( x = -2 \).
- The domain includes all real numbers except \( x = 2 \) and \( x = -2 \).

2. **Horizontal Asymptote**:
- The degrees of the numerator and denominator are both 2 (the highest power of \( x \) in each).
- Since the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients:
\[
y = \frac{2}{1} = 2.
\]

Thus, the domain is all \( x \) except \( x = 2 \) and \( x = -2 \), and the horizontal asymptote is \( y = 2 \).

This approach allows students to systematically find the domain and horizontal asymptote, helping them analyze the key features of rational functions.

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study