Explore BrainMass

Explore BrainMass

    Rational functions, polynomials

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Explain what makes a function a polynomial. Give an example of a function that is a polynomial and a function that is not a polynomial.

    Given the polynomial f(x) = 2x3 + 5x2 - x - 5, answer the following questions:

    Graph the polynomial. What is the degree of the polynomial?
    Explain the constant in the polynomial and how it relates to the graph.
    Define the x-intercepts of the polynomial.
    Explain the relative maximum and relative minimum of the polynomial in the interval [-3,2]. Is there a different between the relative maximum and minimum and the absolute maximum and minimum?
    Part II:

    What is a rational function? Give an example. For your example, answer or perform the following:

    Graph the rational function.
    Give the domain and range of the rational function.
    Find the vertical and horizontal asymptotes of the rational function.

    © BrainMass Inc. brainmass.com December 24, 2021, 8:10 pm ad1c9bdddf

    SOLUTION This solution is FREE courtesy of BrainMass!

    A function f(x) is a polynomial if it has the form

    a_0 + a_1 x + a_2 x^2 + ... + a_n x^n

    where a_0, a_1, a_2, .., a_n are numbers.

    Given the polynomial f(x) = 2x^3 + 5x^2 - x - 5

    (a) visit


    and enter 2x^3 + 5x^2 - x - 5

    in the box

    (b) the polynomial has degree 3, because 3 is the largest exponent on x

    (c) the number at the end (also referred to as the constant coefficient of the polynomial) is precisely the y-intercept
    of the graph (namely, where the graph intersects the y-axis)

    (d) the x-intercepts are the points at which the graph (function) crosses the x-axis

    (e) the relative maximum (resp. minimum) points on some interval are points in this interval whose y-values
    are greatest (resp. smallest)

    (f) a function can have no absolute max/min points, but it may well have relative max/min points; as an example, consider the straight line f(x) = x + 1

    A rational function is a quotient of two polynomial, that is R(x) is rational if R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials, and Q(x) is nonzero for all x.

    An example of a rational function would be 1/(x - 2); another example: (x - 2)/(x - 3)

    Go to the website above and enter (x - 2)/(x - 3)
    to graph this function.

    (a) This function is defined for all x for which the denominator is not zero, i.e. Domain = {x : x =/= 3} (where =/= means not equal to).

    (b) Range = {x : -oo < x < 1} U {x : 1 < x < oo}; that is, all real numbers except 1, i.e. R - {1}.

    (c) It has a vertical asymptote at x = 3; it has a horizontal asymptote at y = 1.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 8:10 pm ad1c9bdddf>