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Math Help - analyzing a function

  1. #1
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    Question analyzing a function

    hi!

    this is the given function:

    f(x) = |x+5|+ (|x^2+5x+6|/x)

    i divided it up into sections:

    f(x) = (x+5)+ (x^2+5x+6)/(x)) when -5<x<-3 x>-2

    f(x)= -(x+5)+ (-(x^2+5x+6)/(x)) when x<-5 -3<x<-2

    i was going to analyze these two functions but I'm confused.
    could someone please help me?!?!?!

    thanks!
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  2. #2
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    Quote Originally Posted by Lilly100 View Post
    hi!

    this is the given function:

    f(x) = |x+5|+ (|x^2+5x+6|/x)

    i divided it up into sections:

    f(x) = (x+5)+ (x^2+5x+6)/(x)) when -5<x<-3 x>-2

    f(x)= -(x+5)+ (-(x^2+5x+6)/(x)) when x<-5 -3<x<-2
    Hello,

    use the definition of an absolute value:

    |x| = \left\{ \begin{array}{lcr}x&if&x\geq 0\\-x&if&x<0 \end{array}\right.

    With your problem you have the constraints:

    |x+5|=  \left\{ \begin{array}{lcr}x+5&if&x\geq -5\\-(x+5)&if&x<-5 \end{array}\right........... and

    |x^2+5x+6| = |(x+2)(x+3)|=  \left\{ \begin{array}{rcl}x^2+5x+6x&if&x\leq -3~\vee~x\geq -2\\-(x^2+5x+6)&if&-3 < x -2 \end{array}\right.

    Additionally you have to observe that x \neq 0

    Your function becomes:
    <br />
f(x)=\left\{ \begin{array}{rcr}-(x+5)+\frac{x^2+5x+6}{x}&if&x <-5 \\  (x+5)+\frac{x^2+5x+6}{x}&if&-5 <x \leq -3 \\ (x+5)+\frac{-(x^2+5x+6)}{x}&if&-3 < x < -2 \\ (x+5)+\frac{x^2+5x+6}{x}&if& x\geq -2 \wedge x\neq 0 \end{array}\right.
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  3. #3
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    thanks for your reply!
    Am I supposed to analyze the four different functions?
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  4. #4
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    Quote Originally Posted by Lilly100 View Post
    thanks for your reply!
    Am I supposed to analyze the four different functions?
    Hi,

    of course!

    you can collect the summands but you have to observe the conditions quite carefully.

    For instance:

    (x+5)+\frac{x^2+5x+6}{x}= \frac{x^2+5x+x^2+5x+6}{x}= \frac{2x^2+10x+6}{x}~,~x \geq-2 ~ \wedge ~ x\neq 0
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