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Math Help - Linear Function

  1. #1
    mlg
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    Linear Function

    I would appreciate help with the following problem please.
    What values of c in the function f : R to R given by f(x) =mx + c, is f linear.
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  2. #2
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    Re: Linear Function

    Quote Originally Posted by mlg View Post
    I would appreciate help with the following problem please.
    What values of c in the function f : R to R given by f(x) =mx + c, is f linear.
    QUESTION: Do you know the definition of linear function?
    If so there is a place to start.
    If not then look it up.
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  3. #3
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    Re: Linear Function

    Any function of the form f(x)= ax+ c is linear for any numbers a and c.
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    Re: Linear Function

    Quote Originally Posted by HallsofIvy View Post
    Any function of the form f(x)= ax+ c is linear for any numbers a and c.
    Actually there are two uses for the term linear function.
    Moreover, this question was posted in the number theory forum which suggests the second meaning to me.
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  5. #5
    mlg
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    Re: Linear Function

    Thank you Plato & Hallsoflvy for the quick reply.
    I will now look up the definition of linear function.
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  6. #6
    mlg
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    Re: Linear Function

    The definition of linear function is:
    A linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:
    f(x + y) = f(x) + f(y)
    f(ax) = af(x)

    How does this definition tie in with the value of c?
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  7. #7
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    Re: Linear Function

    Quote Originally Posted by mlg View Post
    The definition of linear function is:
    A linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:
    f(x + y) = f(x) + f(y)
    f(ax) = af(x)
    How does this definition tie in with the value of c?
    $f(x)=ax+c\\f(y)=ay+c\\f(x+y)=a(x+y)+c\\f(x)+f(y)= ~?$
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  8. #8
    mlg
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    Re: Linear Function

    Thanks again Plato.
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    Re: Linear Function

    To underscore this, I claim for a linear function $f: \Bbb R \to \Bbb R$ we must have:

    $f(0) = 0$.

    By LINEARITY:

    $f(x + y) = f(x) + f(y)$. Taking $x = y = 0$, we get:

    $f(0) = f(0+0) = f(0) + f(0)$ so that:

    $0 = f(0) - f(0) = (f(0) + f(0)) - f(0) = f(0)) + (f(0) - f(0)) = f(0) + 0 = f(0)$.

    Now if $f(x) = mx + c$, what is $f(0)$, and what does that tell you about what $c$ MUST be for $f(x) = mx + c$ to be linear?
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  10. #10
    mlg
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    Re: Linear Function

    Thanks Deveno for your time and effort.
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  11. #11
    mlg
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    Re: Linear Function

    Thanks Plato for your response.
    I made the following attempt:
    f(x) = ax + c
    f(y) = ay + c
    f(x + y) = a(x + y) + c
    f(x) + f(y) = ax + c + ay + c = a(x + y) + 2c.
    But a(x + y) + c is not equal to a(x + y) +2c.
    i.e. c is not equal to 2c, unless c = 0.
    (The question mentioned values, but c = 0 is only one value).
    I would appreciate your comment.
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