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Math Help - Rational functions and their graphs,rational inequalities, transformations

  1. #1
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    Rational functions and their graphs,rational inequalities, transformations

    Hi all,

    I am an army wife taking a distance learning college algebra course. It is condensed- 8 weeks- yikes! And I am having some trouble with a few problems on my midterm review. If you can give me any assistance, I would greatly appreciate it!

    1. Find the domain of (f/g)x when f(x)=6x ^2-3


    and g(x)= x^2-2x-3

    2. Find the domain of the composite function f with g

    f(x)=2/x+7 g(x)=28/x

    3. Given functions f and g, determine the domain of f+g
    f(x)= 4x+5 g(x)=5/x+9

    5. Graph the polynomial function x^4+ 4x^3 +4x^2
    ( you don't have to solve, can you just tell me how i go about graphing it?)

    6. graph the rational function -4x/x+1

    (dont have to solve, but can you give me steps to graphing it)

    Thank you all so much for your help! This online course tells me the problems I got wrong on the midterm review, unfortunately It does not tell me the right answer or how to solve them correctly. God bless all of you math geniuses! I hope I can get there someday
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  2. #2
    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by ashmo100388 View Post
    Hi all,

    I am an army wife taking a distance learning college algebra course. It is condensed- 8 weeks- yikes! And I am having some trouble with a few problems on my midterm review. If you can give me any assistance, I would greatly appreciate it!

    1. Find the domain of (f/g)x when f(x)=6x ^2-3


    and g(x)= x^2-2x-3

    2. Find the domain of the composite function f with g

    f(x)=2/x+7 g(x)=28/x

    3. Given functions f and g, determine the domain of f+g
    f(x)= 4x+5 g(x)=5/x+9

    5. Graph the polynomial function x^4+ 4x^3 +4x^2
    ( you don't have to solve, can you just tell me how i go about graphing it?)

    6. graph the rational function -4x/x+1

    (dont have to solve, but can you give me steps to graphing it)

    Thank you all so much for your help! This online course tells me the problems I got wrong on the midterm review, unfortunately It does not tell me the right answer or how to solve them correctly. God bless all of you math geniuses! I hope I can get there someday
    Hi!

    f(x) = 6x^2 - 3

    g(x) = x^2 - 2x - 3

    (\frac{f}{g})(x) = \frac{6x^2 - 3}{x^2 - 2x - 3}

    The domain of a polynomial function is all real numbers. However, when it is in the form of a rational function, the denominator can not = 0. First factor the denominator.

    (\frac{f}{g})(x) = \frac{6x^2 - 3}{x^2 - 2x - 3}

    = \frac{6x^2 - 3}{(x +1)(x-3)}

    So, the domain is all real numbers except at x = -1 and x = 3

    So in interval notation: (-\infty,-1) U (-1,3) U (3, \infty)

    Back in a sec with the next ones. just didn't want you to have to wait forever!
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  3. #3
    Senior Member mollymcf2009's Avatar
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    #2

    f(g(x)) means that you will put g(x) into f(x) everywhere there is a x. Like this:

    f(x) = \frac{2}{x + 7}

    g(x) = \frac{28}{x}

    f(g(x)) = \frac{2}{(\frac{28}{x} + 7)}

    Get a common denominator:

    = \frac{2}{\frac{28 + 7x}{x}}

    Then multiply the top by the reciprocal of the bottom to get rid of the fraction in the denominator:

    = \frac{2}{1} \cdot \frac{x}{28 + 7x}

    = \frac{2x}{28 + 7x}

    Now to find where the domain is restricted, put the denominator = 0

    7x + 28 = 0

    x = -4

    So, the domain is all real numbers except when x = -4, or (-\infty,-4) U (-4,\infty)
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  4. #4
    Senior Member mollymcf2009's Avatar
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    #3

    (f+g)(x) is the same as f(x) + g(x)

    f(x) = 4x + 5

    <br />
g(x) = \frac{5}{x + 9}

    (f + g) = (4x + 5) + (\frac{5}{x + 9})

    Get a common denominator:

    = \frac{4x (x + 9) + 5 (x + 9) + 5}{x + 9}

    Distribute and collect like terms:

    = \frac{4x^2 + 36x + 5x + 45 + 5}{x + 9}

    = \frac{4x^2 + 41x + 50}{x + 9}

    Now you have a rational polynomial function. Like I showed you in the first ones, put the denominator = 0 and solve for x. This will show you where your domain in restricted. See if you can try it on this one.
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  5. #5
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    Quote Originally Posted by ashmo100388 View Post
    [snip]
    3. Given functions f and g, determine the domain of f+g
    f(x)= 4x+5 g(x)=5/x+9

    5. Graph the polynomial function x^4+ 4x^3 +4x^2
    ( you don't have to solve, can you just tell me how i go about graphing it?)

    6. graph the rational function -4x/x+1

    [snip]
    3. Look at the domain of each function. The domain of f + g is what's common to both f and g ... In this case, all real numbers except x = -9.

    5. Note that x^4 + 4x^3 + 4x^2 = x^2 (x^2 + 4x + 4) = x^2 (x + 2)^2.

    6. Note that \frac{-4x}{x+1} = -4 \left( \frac{x}{x + 1} \right) = -4 \left( \frac{(x + 1) - 1}{x + 1} \right) = -4 \left( 1 - \frac{1}{x + 1} \right) = \frac{4}{x + 1} - 4.

    I hope you recognise it as a rectangular hyperbola.
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