Function on both sides of an inequality?

[phymatter]

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?

 

[jambaugh]

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.

 

[uart]

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

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

 

[jambaugh]

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

 

[Natespank]

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

 

[HallsofIvy]

Are those monotone functions?

 

[Natespank]

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