analyzing a function

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January 5th 2008, 06:41 AM
Lilly100

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!
January 5th 2008, 07:12 AM
earboth

Quote:

Originally Posted by Lilly100

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:
$<br /> f(x)=\left\{ \begin{array}{rcr}-(x+5)+\frac{x^2+5x+6}{x}&if&x <-5 \\ (x+5)+\frac{x^2+5x+6}{x}&if&-5 <x \leq -3 \\ (x+5)+\frac{-(x^2+5x+6)}{x}&if&-3 < x < -2 \\ (x+5)+\frac{x^2+5x+6}{x}&if& x\geq -2 \wedge x\neq 0 \end{array}\right.$
January 5th 2008, 07:32 AM
Lilly100

thanks for your reply!
Am I supposed to analyze the four different functions?
January 5th 2008, 07:44 AM
earboth

Quote:

Originally Posted by Lilly100

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:

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