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Math Help - Algebra 2 problems, need help

  1. #1
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    Algebra 2 problems, need help

    Ok, these are problems from a summer packet I have to turn in for precalc honors. The problems are based around algebra 2.
    Find the inverses, f^-1(x) of the following functions using 3 different methods.

    f(x)=x^2 +2x -5

    f(x)=log_9e^sin(x/2) <---I really need help with this one.


    f(x)=2x+1
    3-x

    Solve each of the following systems algebraically and using matrices
    √3x - 2y = -1

    2x + 5y^2 = 12
    Last edited by Kingreaper; August 14th 2008 at 04:41 PM.
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  2. #2
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    Hello, Kingreaper!

    Do you know how to find an inverse function?


    Find the inverses, f^{\text{-}1}(x) of the following functions
    . . using 3 different methods.
    . . I don't know three different methods!

    f(x)\:=\:x^2 +2x -5

    (1) Replace f(x) with y\!:\;\;y \;=\;x^2+2x-5


    (2) Interchange x's and y's: . x \;=\;y^2 + 2y - 5


    (3) Solve for y

    . . We have: . y^2 + 2x - (x+5) \;=\;0

    . . Quadratic Formula: . y \;=\;\frac{\text{-}2\pm\sqrt{2^2 + 4(x+2)}}{2} \;=\;\frac{\text{-}2 \pm\sqrt{4x+12}}{2}

    . . . . y \;=\;\frac{\text{-}2 \pm2\sqrt{x+3}}{2} \;=\;\text{-}1 \pm\sqrt{x+3}


    (4) Replace y with f^{\text{-}1}(x)\!:\;\;f^{-1}(x) \;=\;-1 \pm\sqrt{x+3}


    Note that this inverse is not truly a function.



    f(x)\:=\:\log_9\!\left[e^{\sin\frac{x}{2}}\right]

    (1)\;\;y \;=\;\log_9\!\left[e^{\sin\frac{x}{2}}\right]

    (2)\;\;x \;=\;\log_9\!\left[e^{\sin\frac{y}{2}}\right]


    (3)\;\;\log_9\!\left[e^{\sin\frac{y}{2}}\right] \;=\;x

    . . . . . . e^{\sin\frac{y}{2}} \;=\;9^x

    . . . . . . \sin\frac{y}{2} \;=\;\ln\left(9^x\right) \;=\;x\!\cdot\!\ln 9

    . . . . . . . . \frac{y}{2} \;=\;\arcsin\left(x\!\cdot\!\ln 9\right)

    . . . . . . . . y \;=\;2\!\cdot\!\arcsin\left(x\!\cdot\!\ln9\right)


    (4)\;\;f^{-1}(x) \;=\;2\!\cdot\!\arcsin\left(x\!\cdot\!\ln9\right)




    f(x) \:=\:\frac{2x+1}{3-x}

    (1)\;\;y \;=\;\frac{2x+1}{3-x}

    (2)\;\;x \;=\;\frac{2y+1}{3-y}

    (3)\;\;\frac{2y+1}{3-y} \:=\:x \quad\Rightarrow\quad 2y+1\:=\:x(3-y) \quad\Rightarrow\quad 2y + 1 \:=\:3x - xy

    . . 2y + xy \:=\:3x - 1 \quad\Rightarrow\quad y(2+x) \:=\:3x-1 \quad\Rightarrow\quad y \:=\:\frac{3x-1}{x+2}


    (4)\;\;f^{-1}(x) \;=\;\frac{3x-1}{x+2}



    Is there a typo in the last problem?
    Matrices work only on linear systems of equations.

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  3. #3
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    Thank you so much, but I have a few more questions. First, how do you type the equations so well, with the powers in the right place without the ^? Second, how do you do the non-linear problem algebraically instead of with matrices? Also, here is another I don't understand.
    Find f times g, f/g, f of g, g of f

    f(x)=6√x + x^2
    5th root (x-2)

    g(x) 3
    5(3rd root(x^2))
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  4. #4
    Junior Member ibnashraf's Avatar
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    Quote Originally Posted by Kingreaper View Post

    f(x)=6√x + x^2
    5th root (x-2)

    g(x) 3
    5(3rd root(x^2))
    hope the following are the equations u referring to above ......

    f(x) \:=\: \frac {6\sqrt x + x^2}{\sqrt[5]{x - 2}} \quad\Rightarrow\quad \frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}}

    g(x) \:=\: \frac {3}{5\sqrt[3]{x^2}} \quad\Rightarrow\quad         \frac {3}{5x^\frac{2}{3}}


    Quote Originally Posted by Kingreaper View Post
    how do you type the equations so well, with the powers in the right place without the ^?
    first time i'm practicing typing out this equation stuff as well ..... there is a forum called 'laTex help' under 'maths help forum lounge'. u may want to check that out to understand what's going on.
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  5. #5
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    yes, those are the equations, lol. Thanks for the latex tip, I'm gonna try that now.
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  6. #6
    Junior Member ibnashraf's Avatar
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    Also, here is another I don't understand.
    Find f times g, f/g, f of g, g of f


    ok i'll try and give this a shot to get some more practice with this laTex programming thing ....


    f(x) \times g(x)  \:=\: \frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} \times \frac {3}{5x^\frac{2}{3}} \quad\Rightarrow\quad \frac {18x^\frac{1}{2} + 3x^2}{5x^\frac{2}{3}(x-2)^\frac{1}{5}} etc

    and

    \frac{f(x)}{g(x)} \:=\: \frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} <br />
\times \frac {5x^\frac{2}{3}}{3} \quad\Rightarrow\quad <br />
\frac {30x^\frac{7}{6}+5x^\frac{8}{3}}{3(x-2)^\frac{1}{5}} etc


    Also;

     fg(x) = f\left[\frac {3}{5x^\frac{2}{3}}\right] <br />
\quad\Rightarrow\quad <br />
\frac{6\left[\frac {3}{5x^\frac{2}{3}}\right]^\frac{1}{2}+\left[\frac {3}{5x^\frac{2}{3}}\right]^2}{\left[\frac {3}{5x^\frac{2}{3}} - 2\right]^\frac{1}{5}}<br /> <br />
etc

    and

     gf(x) = g\left[\frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} \right]<br />
\quad\Rightarrow\quad \frac{3}{5\left[\frac{6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}}\right]^\frac{2}{3}} etc
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  7. #7
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    Thanks, lol here are a couple more, and these are the last ones I need help with on the packet.

    1. Forces of 85 pounds and 50 pounds act on a single point. The angle between the forces is 15 degrees. Find the dirrection and magnitude of the resultant force.

    2. A pet supply company mixes 2 brands of dry dog food. Brand X costs $15 per bag and ciontains 8 units of nutritional element A, 1 unit of nutritional element B, and 2 units of nutritional element C. Brand y costs $30 per bag and contains 2 units of nutritional element A, 1 unit of nutritional element B, and 7 units of nutritional element C. Each bag of mixed dog food must contain at least 16 units of A, 5 units of B, and 20 units of C. Find the number of bags of brands X and Y that should be mixed to produce a mixture meeting the minimum nutritional requirements and having a minimum cost.

    3. 2 planes start from the same airport and fly in opposite directions. The second plane starts 1/2 hour after the first, but travels 80 km faster. Find the air speed of each plane if 2 hours after the first plane departs the planes are 3200 km apart.

    4.2 ships leave port and 9 am. one travels at a bearing of N 53 degrees W at 12 mph and the other travels at a bearing of S 67 degrees W at 16 mph. Approximate how fart apart they are at noon that day.

    5. A 55 gallon barrel contains a mixture with a concentration of 30%. How much of this mixture must be withdrawn and replaced by a 100% concentrate to bring the mixture up to 50% concentration?

    6. Pluto moves in an elliptical orbit with the sun at one of the foci. The lengh of half the major axis is 3.666 x 10^9 miles and the eccentricity is 0.248. Find the smallest and greatest distance of Pluto from the center of the sun.

    7.A right circular cone has a base of radius 1 and a height of 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side length of the cube?

    8. A worked can cover a parking lot with asphalt in 10 hours. With the help of an assisstant, the work can be done in 6 hours. How long would it take the assistant working alone to cover the parking lot?

    Thanks in advance ~ Kingreaper
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