Results 1 to 3 of 3

Math Help - Domains and Inverse functions

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    1

    Post Domains and Inverse functions

    Hi, while I was working on my pre-calc textbook. I came across some questions that I need help on. Showing me how you did it would be more appreciable than giving me the answer. Thank you!

    1) A figure is in the shape of a rectangle surmounted by an equilateral triangle. The figure has perimeter 25m. among all such figures, find length ''x'' of the base of the figure with the largest area. State the domain of your function.
    Domains and Inverse functions-.png

    2)Find the expression for the inverse function f^-1

    f(x)= 1/(x-2)^2 if x>2

    5 + x if x< -5

    3) A cone of height 20m and radius 8m contains half the volume of water that it could contain if it were full. What is the height (or depth) ''h'' of the water?
    Approx.

    Domains and Inverse functions-b.png
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Dec 2009
    Posts
    226
    1) The area of the figure is the sum of the areas of the rectangle and the equilateral triangle. The area of the rectangle (with base x and length y) is given by A = xy. The area of the equilateral triangle is given by A = \dfrac{\sqrt{3}}{4}x^2.

    The sum of these areas is given by A = xy + \dfrac{\sqrt{3}}{4}x^2.

    The perimeter of the figure is given by 25 = 2y + 3x. Therefore, y = \dfrac{25 - 3x}{2}.

    Now, substitute y = \dfrac{25 - 3x}{2} in the area formula.

    A = x\dfrac{25 - 3x}{2} + \dfrac{\sqrt{3}}{4}x^2 = \dfrac{25}{2}x - \dfrac{3}{2}x^2 + \dfrac{\sqrt{3}}{4}x^2

    Thus, the area function is a parabola whose maximum will occur at its vertex. Its its vertex can be found either graphically or algebraically.

    2) f(x) is a piece-wise function, so tackle the problem piece by piece. Start with the first piece:

    f(x) = \dfrac{1}{\left(x-2\right)^2}

    Find its domain and range since they correspond to the range and domain of the inverse function, respectively.

    Here is the domain of f(x) (I'll let you find its range): (2,\infty)

    Then, find the inverse of y = \dfrac{1}{\left(x-2\right)^2} by switching y and x and solving for the "new" y.

    Repeat the steps for the second piece: f(x) = 5 + x

    Here is the final answer:

    Spoiler:
    f(x)=\begin{cases}-2 + \sqrt{x},\quad& (0,\infty)\\<br />
x - 5_,\quad& (-\infty,0)\end{cases}


    3) The problem mentions the volume of a cone, so start with the formula for the volume of a cone:

    V = \dfrac{1}{3}\pi r^2 h

    You know the radius and height of the cone, so substitute these values into the volume formula.

    V = \dfrac{1}{3}\pi \cdot 8^2 \cdot 20 = \dfrac{1280\pi}{3}

    But the volume of the cone is only half the volume if it were full, so V = \dfrac{640\pi}{3}.

    The radius of the water is still 8 meters, and you just found its volume to be \dfrac{640\pi}{3} cubic meters. Use the volume formula to solve for the height h.
    Last edited by NOX Andrew; February 27th 2011 at 07:25 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,110
    Thanks
    982
    (1) base of rectangle = x

    height of rectangle = \dfrac{25-3x}{2}

    A = \dfrac{x(25-3x)}{2} + \dfrac{\sqrt{3} \cdot x^2}{4}

    note that x > 0 and 25-3x > 0
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inverse trig functions and implied domains
    Posted in the Trigonometry Forum
    Replies: 5
    Last Post: August 14th 2011, 05:10 AM
  2. [SOLVED] Domains with e and inverse functions
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: March 6th 2010, 04:01 AM
  3. functions and domains
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: September 1st 2008, 08:04 AM
  4. functions and domains
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: August 31st 2008, 10:07 PM
  5. Domains of functions
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 28th 2008, 08:41 AM

Search Tags


/mathhelpforum @mathhelpforum