Question

Make up a rational function that has \(y=2 x+1\) as an oblique asymptote.

Short answer

The rational function can be \(\frac{2x^2 + x + c}{x}\).

Step-by-step solution

Understand the Concept of Oblique Asymptote

The function must approach the line \(y = 2x + 1\) as \(x\) goes to infinity. This occurs when the degree of the numerator is one more than the degree of the denominator.

Formulate the Rational Function

A rational function \(R(x)\) that approaches \(y = 2x + 1\) can be written as \(R(x) = \frac{p(x)}{q(x)}\) where \( \text{deg}(p(x)) = \text{deg}(q(x)) + 1 \). We can choose \(p(x) = 2x^2 + x + c\) and \(q(x) = x\).

Divide the Polynomials

Divide the polynomials to ensure the asymptote is correct. Using long division: \(\frac{2x^2 + x + c}{x} = 2x + 1 + \frac{c}{x}\). The remainder term \(\frac{c}{x}\) becomes negligible as \(x\) approaches infinity.

Verify the Oblique Asymptote

As \(x\) approaches infinity, the function \(\frac{2x^2 + x + c}{x}\) simplifies to \(2x + 1\), confirming that the oblique asymptote is \(y = 2x + 1\).

Conclusion

Therefore, a rational function with an oblique asymptote \(y = 2x + 1\) can be: \(\frac{2x^2 + x + c}{x}\).

Key concepts

Rational Function

A rational function is a type of function that is expressed as the ratio of two polynomials. In mathematical terms, it is written as: \[ R(x) = \frac{p(x)}{q(x)} \]
where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \). These functions are important because they can show complex behaviors, such as asymptotes and intersections with other lines.

Rational functions can be used to model real-world scenarios like physics problems, engineering designs, and economic trends.

When analyzing a rational function, it's crucial to understand how the degrees of the numerator and the denominator polynomials affect the overall shape and behavior of the function.

Polynomial Division

Polynomial division is an essential tool for simplifying rational functions. It involves dividing one polynomial by another to find a quotient and a remainder. This is similar to how long division works with numbers.

In the context of rational functions, polynomial division helps us find out the asymptotes of the function. Specifically, to identify an oblique asymptote, we perform long division on the polynomials and focus on the quotient (ignoring the remainder for large values of \( x \)).

For example, if we start with \( \frac{2x^2 + x + c}{x} \), polynomial division yields:
\[ 2x + 1 + \frac{c}{x} \]
As \( x \) approaches infinity, the term \( \frac{c}{x} \) becomes small and insignificant, leaving us with the oblique asymptote \( y = 2x + 1 \).

• An important concept: The degree of the numerator must be one more than the degree of the denominator for an oblique asymptote to occur.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial \( 5x^3 + 3x^2 + x + 7 \), the degree is 3 because the highest exponent of \( x \) is 3.

Knowing the degree of the polynomials in a rational function is key to determining the nature of its asymptotes.

• If the degree of the numerator is higher than that of the denominator by exactly one, the function will have an oblique asymptote.
• If the degrees are the same, the function will have a horizontal asymptote.
• If the degree of the denominator is higher, the function will approach zero.

If we want a rational function with an oblique asymptote \( y = 2x + 1 \), we need to ensure that the degree of the numerator is exactly one higher than the denominator. For instance, with a numerator like \( 2x^2 + x + c \) and a denominator like \( x \), the degrees are aligned to give us the desired asymptote.