Math (Functions)

Discussion in 'Community Discussion' started by waloshin, Mar 30, 2009.

  1. waloshin macrumors 68040

    waloshin

    Joined:
    Oct 9, 2008
    #1
    Write the equation of a line which staisfies:

    1) Perpendicular to the line y= 1/3x +5 and x intercept of 2.

    2) I dont understand what this means: For f(x) = 2x^2 -x and g(x) = 3x + 6

    example question : f(f(x)) or f(x) -g(x)

    * For the functions listed in the example questions find the x such that f(x) = g(x).
     
  2. JohnDoe8450 macrumors member

    Joined:
    Feb 3, 2009
    #2
    Are these 2 seperate questions? If so...

    1)
    We have :
    y = 1/3 x +5
    From this, we know that the slope of the line is 1/3 (slope is defined as y variation over x variation, and for each time x increases by 1, y increases by 1/3)
    We can therefore say that the slope of a line perpendicular to this one will be 3. (If 2 lines are perpendicular, their slopes are inverses so slope = 1 / (1/3) = 3 )
    We therefore have :
    y = 3 x + b​

    To find b, we use the fact that this line intercepts x at 2. This means it goes through point x = 2, y = 0. This is also expressed as (2, 0). If we put these values in the equation...
    0 = 3 * 2 + b
    0 = 6 + b
    b = -6​

    We have that the equation of the line which we were asked to find is :
    y = 3 x = 6.​



    2)
    When they say f(x) - b(x) here's what they want : (I used D(x) in a totally arbitrary manner)
    D(x) = f(x) - b(x)
    Then all we do is substitute in the values of f(x) and b(x) and simplify...
    D(x) = ( 2x^2 - x ) - ( 3x + 6 )
    D(x) = 2x^2 - x - 3x + 6
    D(x) = 2x^2 - 4x + 6
    D(x) = x^2 - 2x + 3

    If they ask for f( f(x) ) then here's how that works...
    f(x) basically means f is a function of x. However, x doesn't need to be an integer. x can be anything, I could set x as being 4*u if I wanted to, all I'd do is substitute x = 4u into f(x) so I'd get something like this :
    f(4u) = 2(4u)^2 - 4u
    f(4u) = 32u^2 - 4u
    So, f(f(x))... let's rewrite that as f(F(x)) just so you know which f/F I'm talking about :p They basically want me to substitute x=F(x) into f(x) so we get :
    f(f(x)) = 2(2x^2 - x)^2 - (2x^2 - x)​

    All I've done here is swap out the x's out of 2x^2 - x and put (2x^2 - x) instead. Then, we simplify...
    f(f(x)) = 2( 4x^4 - 4x^3 + x^2) - 2x^2 - x
    f(f(x)) = 8x^4 - 4x^3 + 2x^2 - 2x^2 - x
    f(f(x)) = 8x^4 - 4x^3 - x
    f(f(x)) = x * ( 8x^3 - 4x^2 - 1 )
     
  3. fireshot91 macrumors 601

    fireshot91

    Joined:
    Jul 31, 2008
    Location:
    Northern VA
    #3


    I thought it was the negative reciprocal?
     
  4. Doctor Q Administrator

    Doctor Q

    Staff Member

    Joined:
    Sep 19, 2002
    Location:
    Los Angeles
    #4
    That's correct. The equation for the perpendicular line must have the form
    y = -3 x + b​
    From there you can use the method JohnDoe8450 shows to find b such that the equation is true for x = 2, y = 0:
    y = -3 x + b
    0 = -3 (2) + b
    0 = -6 + b
    b = 6​
    Result:
    y = -3 x + 6​
     

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