Function on both sides of an inequality?

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    Function Inequality
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Discussion Overview

The discussion revolves around the application of functions on both sides of an inequality, exploring whether any function can be applied without checking conditions. The scope includes theoretical considerations of monotonicity and the implications for inequalities.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that not all functions can be applied to both sides of an inequality without checking conditions, citing examples where inequalities do not hold after applying certain functions.
  • Monotone functions are discussed, with participants noting that increasing functions preserve inequalities while decreasing functions reverse them.
  • Examples of monotone increasing functions include adding a constant, multiplying by a positive number, and certain logarithmic and exponential functions.
  • Monotone decreasing functions include the reciprocal and negative multiples, but the reciprocal is noted to be monotone decreasing only when restricted to positive numbers.
  • One participant suggests that for complex inequalities, it is useful to rearrange terms to form f(x) > 0 or f(x) < 0, and to identify boundary points where the inequality may change.
  • A question is raised about the validity of raising both sides of an inequality to a power or taking roots, with uncertainty expressed regarding whether these operations are monotone functions.

Areas of Agreement / Disagreement

Participants generally disagree on the conditions under which functions can be applied to both sides of an inequality, with multiple competing views on the nature of monotonicity and the implications for different types of functions.

Contextual Notes

There are limitations regarding the assumptions about the domains of certain functions, particularly the reciprocal function, and the need to consider discontinuities and undefined points when analyzing inequalities.

phymatter
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Function on both sides of an inequality?

Can we apply any function on both sides of an inequality irrespective of its nature and without checking any conditions?
 
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phymatter said:
Can we apply any function on both sides of an inequality irrespective of its nature and without checking any conditions?
No...not even close.

Note 3 > -4 but (3)^2 < (-4)^2.

You can apply monotone functions which are by definitions which preserve the inequality. A monotone (increasing) function f has the property x>y iff f(x) > f(y). You can also apply a monotone decreasing function which will reverse the inequality.

Examples of monotone increasing functions are adding a constant, multiplying by a positive number, logarithms (where defined), exponentials with base > 1.

Monotone decreasing: reciprocal, negative multiples, exponentials with base < 1.

For general functions you need to identify regions where it is monotone increasing or monotone decreasing.
 


jambaugh said:
Monotone decreasing: reciprocal, negative multiples, exponentials with base < 1.

Careful. Reciprocal is only monotone decreasing if you restrict the domain to positive numbers only.
 


uart said:
Careful. Reciprocal is only monotone decreasing if you restrict the domain to positive numbers only.

Oops, right it monotone decreasing except discontinuous at 0.

phymatter,
If you have a more complicated inequality you should bring all terms to one side so it is of the form f(x) >0 or f(x)<0 or one of the "or equal" cases.

You then find the solution to the equation f(x)=0 and these points are boundaries on regions of the real number line where the inequality can change from being true to false. But you must also consider points where f(x) is discontinuous or undefined. Once you determine all boundary points you can test the inequality between them and it will be either uniformly true or uniformly false throughout those regions.

Think of it as figuring where the graph of the function is above and where it is below the x axis.
 


What about raising both sides to a power, or taking a root of each side?
 


Are those monotone functions?
 


I'm not sure. I think they are sometimes, but not always.
 

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