Ch 5.2 HW
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
