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Thread: Get everything from the graph

  1. #1
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    Smile Get everything from the graph

    $\displaystyle \mathfrak{hello}$
    i'm trying to get all the possible informations about a function from its graph and i mean by informations;limits,asymptotes..
    i hope someone would help me extract those informations from this example :
    Let : $\displaystyle f : \mathbb{R}\to \mathbb{R}\setminus \left \{ -1 \right \}$
    and here's its graph :
    Attached Thumbnails Attached Thumbnails Get everything from the graph-graph.png  
    Last edited by Raoh; Jun 30th 2009 at 12:53 PM.
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  2. #2
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    Hello, Raoh!

    I'm trying to get all the possible informations about a function from its graph
    and i mean by informations: limits, asymptotes . . .
    It appears to have a vertical asymptote: $\displaystyle x = -1$
    . . and a horizontal asymptote: $\displaystyle y = 1$

    It also seems to be a hyperbola.


    The hyperbola: .$\displaystyle y \:=\:\frac{a}{x}$ .has center (0,0) and appears in Quadrants 1 and 3.
    The hyperbola: .$\displaystyle y \:=\:-\frac{a}{x}$ .has center (0,0) and appears in Quadrants 2 and 4.


    Our hyperbola has center (-1, 1) and seems to contain: (3,0), (1,-1), (0,-3).

    I would conclude that the function is: .$\displaystyle y - 1 \:=\:\frac{-4}{x+1} \quad\Rightarrow\quad\boxed{ y \;=\;\frac{x-3}{x+1}}$

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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by Raoh View Post
    $\displaystyle \mathfrak{hello}$
    i'm trying to get all the possible informations about a function from its graph and i mean by informations;limits,asymptotes..
    i hope someone would help me extract those informations from this example :
    Let : $\displaystyle f : \mathbb{R}\to \mathbb{R}\setminus \left \{ -1,3 \right \}$
    and here's its graph :
    Hello : those informations from this example is:
    f(0)=-3
    f(3)=0
    limf:infini (x--> -1)
    limf =1 (x---->infini)
    Last edited by dhiab; Jun 30th 2009 at 11:43 AM.
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  4. #4
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    Smile

    thanks guys,but i still don't understand
    i'm gonna try to illustrate what i wanted to understand from that graph
    $\displaystyle \lim_{x\to +\infty }f(x) = ?$
    $\displaystyle \lim_{x\to -\infty }f(x) = ?$
    and i don't need just values,i wanna know how
    thanks a lot.
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  5. #5
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Raoh View Post
    thanks guys,but i still don't understand
    i'm gonna try to illustrate what i wanted to understand from that graph
    $\displaystyle \lim_{x\to +\infty }f(x) = ?$
    $\displaystyle \lim_{x\to -\infty }f(x) = ?$
    and i don't need just values,i wanna know how
    thanks a lot.
    If $\displaystyle \lim_{x\to\infty}f(x)=L$ then the line $\displaystyle y=L$ is a horizontal asymptote of the graph of $\displaystyle f$.

    In your case, it appears from the graph that you have provided, that a good approximation for $\displaystyle L$ is $\displaystyle \frac{1}{2}$
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  6. #6
    Super Member dhiab's Avatar
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    Quote Originally Posted by Raoh View Post
    thanks guys,but i still don't understand
    i'm gonna try to illustrate what i wanted to understand from that graph
    $\displaystyle \lim_{x\to +\infty }f(x) = ?$
    $\displaystyle \lim_{x\to -\infty }f(x) = ?$
    and i don't need just values,i wanna know how
    thanks a lot.
    Hello : I'have anathor solution
    That function is homographic :
    1)

    2)

    3 )
    x=-1 is asymtot :
    4 ) y=1 is asymtot :

    I'have this system :
    Subst in f(x) :

    conclusion :
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  7. #7
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    Smile

    thanks again for everyone,but i should say that the last post got me a little twisted,because i can't tell from the graph that the function should be written like this $\displaystyle f(x) = \frac{ax+b}{cx+d}$ ,in contrary i like how "Soroban" concluded that the function is indeed $\displaystyle f(x) = \frac{x-3}{x+1}$,but i don't understand his steps , any help ?
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  8. #8
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Raoh View Post
    thanks again for everyone,but i should say that the last post got me a little twisted,because i can't tell from the graph that the function should be written like this $\displaystyle f(x) = \frac{ax+b}{cx+d}$ ,in contrary i like how "Soroban" concluded that the function is indeed $\displaystyle f(x) = \frac{x-3}{x+1}$,but i don't understand his steps , any help ?
    Where the lines $\displaystyle y=1$ and $\displaystyle x=-1$ intersect is the center of the hyperbola.
    $\displaystyle x+1$ must appear irreducibly in the denominator because the function is undefined at $\displaystyle x=-1$.

    He also made use of the intercepts.
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  9. #9
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    Smile

    alright,the domain can tell what should be written in the denominator(i'm not sure if always),what about the numerator ?
    thanks.
    sorry to bother you guys.
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