Chapter 8.3, Problem 12E

Solution

To find: The asymptotes for the function $f\left(x\right)=\frac{6x^2+14x+7}{2x+3}$ .

The oblique asymptote of the function is $y=3x+\frac{5}{2}$ .

Given:

The rational function is $f\left(x\right)=\frac{6x^2+14x+7}{2x+3}$ .

Concept used:

If the degree of numerator is one greater than the degree of denominator where there is no common factor in numerator and denominator, then the rational function has oblique asymptote which can be obtained by dividing the numerator by denominator where quotient obtained is the oblique asymptote.

Calculation:

On dividing the numerator by denominator for the rational function $f\left(x\right)=\frac{6x^2+14x+7}{2x+3}$ .

  

From the obtained division, it can be interpreted that the quotient of the division is $3x+\frac{5}{2}$ . Therefore, the function has oblique asymptote $y=3x+\frac{5}{2}$ .

Now, the graph of the function $f\left(x\right)=\frac{6x^2+14x+7}{2x+3}$ and the oblique asymptote $y=3x+\frac{5}{2}$ is shown below where the oblique asymptote $y=3x+\frac{5}{2}$ is a line which passes through the origin $\left(0,2.5\right)$ .

  

Conclusion:

From the graph of the function, it can be interpreted that the oblique asymptote of the function is $y=3x+\frac{5}{2}$ .