What type of function can I apply to both sides of an inequality in order to obtain an equivalent inequality? I'd like to know what are the types of functions that I can apply to both sides of an inequality in order to get an equivalent inequality. Are those functions the ones that are strictly increasing or strictly decreasing? I'm not sure about it.
Also, I'd like to get a recommendation about good sources where I can read about it.
Thanks.
If by "equivalent inequality" you mean $a&lt;b \iff f(a)&lt;f(b)$ then $f$ must be strictly increasing - for example:

$a&lt;b \iff a+k &lt; b+k\\
a&lt;b \iff ka &lt; kb, \space k&gt;0\\
a&lt;b \iff a^3 &lt; b^3\\
a&lt;b \iff \root{3}\of{a} &lt; \root{3}\of b\\
a&lt;b \iff \tan(a) &lt; \tan(b), \space -\frac \pi 2 &lt;a,b&lt;\frac \pi 2$

If $f$ is strictly decreasing then the inequality is reversed i.e. the equivalence becomes $a&lt;b \iff f(a) &gt; f(b)$. For example:

$a&lt;b \iff ka &gt; kb, \space k&lt;0\\
a&lt;b \iff \frac 1 a &gt; \frac 1 b, \space a,b&gt;0$
It is always true that $a=b$ implies $f(a)=f(b)$, no matter what function $f$ is.
But in order to justify that $f(a)=f(b)$ implies $a=b$ (or, taken together, that $a=b$ is equivalent to $f(a)=f(b)$), you better consider only injective functions $f$.

Indeed, $f(a)=f(b)\implies a=b$ for all applicable $a,b$ is precisely the definition of $f$ being injective. Of course, strictly increasing (or decreasing) functions are in particular injective.