Question

choose whether the statement is true or false: the rational equation y = \frac{3x^{2}+4x + 1}{3x + 4} has a slant asymptote true; the degree of the numerator is one greater than the degree of the denominator false; the degree of the numerator is not greater than the degree of the denominator

Explanation:

Step1: Identify numerator and denominator degrees

For $y = \frac{3x^{2}+4x + 1}{3x + 4}$, the degree of the numerator $3x^{2}+4x + 1$ is 2 (highest - power of $x$ is 2), and the degree of the denominator $3x + 4$ is 1 (highest - power of $x$ is 1).

Step2: Recall slant - asymptote condition

A rational function $y=\frac{f(x)}{g(x)}$ has a slant asymptote when the degree of the numerator $f(x)$ is exactly one greater than the degree of the denominator $g(x)$. Here, $2=1 + 1$.

Answer:

True; the degree of the numerator is one greater than the degree of the denominator