How do increasing functions preserve the direction of the inequality

Question

Say , we have $x < 7$. Then, I read in a book that since the square root function is an increasing function, it preserves the direction of the inequality. This was followed in the book by $\sqrt{x} < \sqrt{7}$. I do not fully understand what it means when the book said the statement in italics above?

What is the intuition behind this; why is this true, in general, for increasing functions? I am asking more for something intuitive than some kind of a proof. Also, can someone please provide an example of a decreasing function (and/or possibly some supporting graphs) that does not preserve the direction of the inequality? Thanks!

Answer 1

By definition an increasing function means

• if $x\le y,$ then $f(x)\le f(y).$
• In other words, $f(x)$ increases as $x$ increases.

Therefore the orientation of the inequality does not change by the function $f.$ For the same reasoning decreasing functions reverse inequalities.

Answer 2

We seem to be dealing with positive numbers here, since you are considering the square root.

Take for example the decreasing function $$f(x)=\frac1x$$

which has the derivative $-\frac1{x^2}$, which is always negative.

Then $$x < 7 \implies \frac1x > \frac17$$ is an example of reversing the inequality. If $x=3$ we can say $\frac13 > \frac17$ because we are dealing with a decreasing function