Questions: Question Which of the following functions will have an oblique asymptote? Select the correct answer below: f(x) = (-2x - 1)(2x + 1) / (5x^2 + x - 4) g(x) = (2 - 8x) / ((3 - 2x)(2x + 3)) h(x) = (-2x - 1)(x^2 - 2) / (2x^2 + 1) p(x) = (1 - 3x) / ((-x - 2)(2x + 2))

Question
Which of the following functions will have an oblique asymptote?

Select the correct answer below:
f(x) = (-2x - 1)(2x + 1) / (5x^2 + x - 4)
g(x) = (2 - 8x) / ((3 - 2x)(2x + 3))
h(x) = (-2x - 1)(x^2 - 2) / (2x^2 + 1)
p(x) = (1 - 3x) / ((-x - 2)(2x + 2))

Transcript text: Question Which of the following functions will have an oblique asymptote? Select the correct answer below: $f(x)=\frac{(-2 x-1)(2 x+1)}{5 x^{2}+x-4}$ $g(x)=\frac{2-8 x}{(3-2 x)(2 x+3)}$ $h(x)=\frac{(-2 x-1)\left(x^{2}-2\right)}{2 x^{2}+1}$ $p(x)=\frac{1-3 x}{(-x-2)(2 x+2)}$

Solution

Solution Steps

To determine which function has an oblique asymptote, we need to check the degrees of the numerator and the denominator of each function. An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator.

Step 1: Analyze the Functions

We need to determine the degrees of the numerators and denominators for each function to identify if they have an oblique asymptote. An oblique asymptote exists if the degree of the numerator is exactly one more than the degree of the denominator.

Step 2: Calculate Degrees

For $ f(x) = \frac{(-2x - 1)(2x + 1)}{5x^2 + x - 4} $:
- Degree of numerator: $ 2 $
- Degree of denominator: $ 2 $
For $ g(x) = \frac{2 - 8x}{(3 - 2x)(2x + 3)} $:
- Degree of numerator: $ 1 $
- Degree of denominator: $ 2 $
For $ h(x) = \frac{(-2x - 1)(x^2 - 2)}{2x^2 + 1} $:
- Degree of numerator: $ 3 $
- Degree of denominator: $ 2 $
For $ p(x) = \frac{1 - 3x}{(-x - 2)(2x + 2)} $:
- Degree of numerator: $ 1 $
- Degree of denominator: $ 2 $