Linear Dependence of Functions with Absolute Value

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Homework Help Overview

The discussion revolves around determining the linear dependence or independence of two functions: f(x) = x^3 and g(x) = x^2|x|. Participants are exploring the implications of the absolute value in the context of linear dependence.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • One participant attempts to derive a relationship between the functions using their derivatives, while another questions the necessity of taking derivatives in this context. There is also a discussion about the behavior of the functions across different values of x, particularly regarding their positivity and implications for linear dependence.

Discussion Status

The discussion is active, with participants offering differing viewpoints on the linear dependence of the functions. Some are questioning assumptions about the functions' behavior, while others are providing reasoning based on the properties of the functions.

Contextual Notes

There is a mention of the functions' positivity and the impact of the absolute value on their relationship. The discussion also hints at the importance of the domain over which the functions are defined.

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Homework Statement


Determine whether the pairs of functions are linear dependent or linearly independent.

f(x) = x^3
g(x) = x2|x|

Homework Equations





The Attempt at a Solution



g(x)=x^2|x| = x^2*sqrt[x^2] = sqrt[x^6] = x^3

f'=3x^2
g'=3x^2

fg'-f'g = 0

Linearly Dependent according to me, Linearly independent according to the book.

I assume it has to do with the absolute value, could someone enlighten me?
 
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Should g'(x) be 2x|x|+x^2(|x|/x) ?
 
note that f is positive for some values of x and negative for others whereas g is always positive. if these two were linearly dependent one would be a constant multiple of the other FOR ALL VALUES OF x. can we multiply a fully positive function by a number so that part of it becomes negative? no. therefore no linear dependence. note also this depends on the set over which g and f are defined.
 
why are you take the derivative?
 

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