Isa Trevino

2020-12-28

The vertical, horizontal and oblique asymptotes of the rational function $R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$ .

Clelioo

Skilled2020-12-29Added 88 answers

Given:

The rational function is defined by$R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$ .

Result used:

Vertical asymptote:

Consider a rational function$R(x)=\frac{p(x)}{q(x)}$ , in its lowest terms. If ris a real zero of the polynomial q(x), then x = r is a vertical asymptote of the function R.

Horizontal or oblique asymptotes:

Consider a rational function$R(x)=p\frac{x}{q}(x)=\frac{{a}_{n}{x}^{n}+{1}_{n-1}{x}^{n-1}+...{a}_{1}x+{a}_{0}}{{b}_{m}}{x}^{+}{b}_{m-1}{x}^{m-1}+...{b}_{1}x+{b}_{0}$ where nis the degree of the polynomial p(x) and mis the degree of the polynomial q(x). If $n=m+1$ , the rational function R has no horizontal asymptote and only has an oblique asymptote given by $y=ax+b$ where $y=ax+b$ is quotient obtained by the polynomial division $\frac{p(x)}{q(x)}$ .

Calculation:

Rewrite the rational function$R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$ as follows.

$R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$

$=\frac{6{x}^{2}+21x-2-7}{3x-1}$

$=3x(2x+7)-1\frac{2x+7}{3x-1}$

$=\frac{(2x+7)(3x-1)}{3x-1}$

$=2x+7$

Hence, the rational function$R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$ in its lowest terms is $R(x)=2x+7$ .

The polynomial in the denominator of$R(x)=2x+7$ is 1 and 1 has no zeros.

Therefore, the rational function$R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$ has no vertical asymptote.

Note that, the degree of the polynomial in the numerator is 2 and the degree of the polynomial in the denominator is 1.

That is, the degree of the polynomial in the numerator is 1 more than the degree of the polynomial in the denominator.

Therefore, the rational function$R(x)=\frac{6{x}^{2}+19x-7}{3x-1}$ has no horizontal asymptote.

The polynomial division$\frac{6{x}^{2}+19x-7}{3x-1}$ gives the quotient as 2x +7.

Therefore, the rational function$\frac{6{x}^{2}+19x-7}{3x-1}$ has an oblique asymptote given by $y=2x+7$ .

Hence, the rational function$\frac{6{x}^{2}+19x-7}{3x-1}$ has no vertical asymptote, no horizontal ext and oblique asymptote is $y=2x$ .

The rational function is defined by

Result used:

Vertical asymptote:

Consider a rational function

Horizontal or oblique asymptotes:

Consider a rational function

Calculation:

Rewrite the rational function

Hence, the rational function

The polynomial in the denominator of

Therefore, the rational function

Note that, the degree of the polynomial in the numerator is 2 and the degree of the polynomial in the denominator is 1.

That is, the degree of the polynomial in the numerator is 1 more than the degree of the polynomial in the denominator.

Therefore, the rational function

The polynomial division

Therefore, the rational function

Hence, the rational function

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