How do increasing functions preserve the direction of the inequality

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!

• The statement in italics is the definition of an increasing function. May 3, 2020 at 23:46

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.

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