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Linear function problem alternative solution

  1. Sep 2, 2014 #1

    Rectifier

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    Hey!
    I am searching for an alternative way to solve this problem
    1. The problem
    For a linear funtion following applies:
    ##f(a+1)=a+2##
    Write the linear function on the ## y=kx+m ## form

    2. Relevant equations
    ##f(a+1)=a+2 \\ y=kx+m ##

    3. The attempt
    Here is how I solved the problem:
    ##f(a+1)=k(a+1)+m \\ k(a+1)+m = a+2 \\ k(a+1)+m = a+1+1 \\ k(a+1)+m = 1 \cdot (a+1)+1 ##
    Following gives ##k=1## and ##m=1##. This means that the equation is:
    $$y=x+1$$

    Any ideas on how I could solve this differently?
     
  2. jcsd
  3. Sep 2, 2014 #2

    Zondrina

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    It is worth noticing ##f(1) = 2, \space f(2) = 3, \space f(3) = 4, ...##

    You can easily find the slope in this manner:

    ##m = \frac{\Delta y}{\Delta x} = 1##

    Then using any one of those points, say ##f(1) = 2##, you find that ##y = mx + b \Rightarrow b = 1##.
     
  4. Sep 2, 2014 #3

    Ray Vickson

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    Much quicker: for any given ##x## let ##a = x+1##. Then the equation says ##f(x) = x+1##.
     
  5. Sep 2, 2014 #4

    Rectifier

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    Yeah that an alternative too. Thanks for the comment :)
     
  6. Sep 2, 2014 #5

    Rectifier

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    Thank you for the comment.

    Do you mean like that:

    ##f((x+1)+1)=(x+1)+2##

    ?

    If yes, whats is the next step?
     
  7. Sep 2, 2014 #6

    Ray Vickson

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    Sorry: there was a typo in my post. It should have said that for any given ##x##, let ##a = x-1##. That gives ##f(x) = x+1##. Since this holds for any ##x## that's it: there is no more to be done.
     
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