Ch 5.2 HW

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Which type of​ asymptote, when it​ occurs, describes the behavior of a graph when x is close to some​ number?

vertical (f R denotes a function and​ if, as x approaches some number​ c, the values StartAbsoluteValue Upper R left parenthesis x right parenthesis EndAbsoluteValue right arrow infinityR(x)→∞ ​[that is, Upper R left parenthesis x right parenthesis right arrow minus infinityR(x)→−∞ or Upper R left parenthesis x right parenthesis right arrow infinityR(x)→∞​], then the line xequals=c is a vertical asymptote of the graph of R.​ If, as x right arrow infinityx→∞ or as x right arrow minus infinityx→−∞​, the values of Upper R left parenthesis x right parenthesisR(x) approach some fixed number​ L, then the line yequals=L is a horizontal asymptote of the graph of R.​ If, as x right arrow infinityx→∞ or as x right arrow minus infinityx→−∞​, the values of Upper R left parenthesis x right parenthesisR(x) approach a linear expression axplus+​b, a not equals 0a≠0​, then the line yequals=axplus+​b, a not equals 0a≠0​, is an oblique​ (or slant) asymptote of the graph of R.)

If, as x approaches some number​ c, the values of ​|R(x)|right arrow→infinity∞​, then the line x=c is a​ _______ _______ of the graph of R.

vertical asymptote

If a rational function is​ proper, then​ _______ is a horizontal asymptote.

y=0 (When a rational function​ R(x) is​ proper, the degree of the numerator is less than the degree of the​ denominator; as xright arrow→−∞ or as x→∞​, the value of​ R(x) approaches 0.)

The domain of every rational function is the set of all real numbers.

False

The graph of a rational function may intersect a vertical asymptote.

False. (The graph of a function will never intersect a vertical asymptote. Note that the graph of a function may intersect a horizontal asymptote.)

For the function F(x)=1/(x−4)^2​, ​(a) graph the rational function using​ transformations, (b) use the final graph to find the domain and​ range, and​ (c) use the final graph to list any​ vertical, horizontal, or oblique asymptotes.

The domain of the given function is StartSet x|x is a real number comma x not equals nothing |x|x is a real number, x≠4.

What are the quotient and remainder when 3 x Superscript 4 Baseline minus x squared3x4−x2 is divided by x cubed minus x squared plus 2x3−x2+2​?

The quotient is 3 x plus 33x+3 and the remainder is 2x^2 -6x-6.

The graph of a rational function may intersect a horizontal asymptote.

True

The quotient of two polynomial expressions is a rational expression.

True

If the degree of the numerator of a rational function equals the degree of the​ denominator, then the ratio of the leading coefficients gives rise to the horizontal asymptote.

True. (If the degree of the numerator of a rational function equals the degree of the​ denominator, then the rational function is improper. Use long division to write the rational function as the sum of a polynomial​ f(x) (the​ quotient) plus a proper rational function StartFraction r left parenthesis x right parenthesis Over q left parenthesis x right parenthesis EndFraction r(x) q(x) ​(r(x) is a​ remainder). If nequals=m ​(the degree of the numerator equals the degree of the​ denominator), the quotient obtained will be the number StartFraction a Subscript n Over b Subscript m EndFraction an bm​, and the line yequals=StartFraction a Subscript n Over b Subscript m EndFraction an bm is a horizontal asymptote.)

If, as x right arrow→ minus−infinity∞ or as x right arrow→ infinity∞​, the values of​ R(x) approach some fixed number​ L, then the line y=L is a​ _________ of the graph of R.

horizontal asymptote

If ​R(x)equals=StartFraction p left parenthesis x right parenthesis Over q left parenthesis x right parenthesis EndFraction p(x) q(x) is a rational function and if p and q have no common​ factors, then R is ______________________.

in lowest terms (Note that the rational expression is in its lowest terms when there is no common factor in the numerator and the denominator.)

For a rational function​ R, if the degree of the numerator is less than the degree of the​ denominator, then R is​ ______.

proper


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