 asigurato7

2022-07-19

Is a horizontal line an increasing or decreasing function?
This is the definition of an increasing and decreasing function.
"A function f(x) increases on an interval I if $f\left(b\right)\ge f\left(a\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }b>a$, where $a,b\in I$. If $f\left(b\right)>f\left(a\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}b>a$, the function is said to be strictly increasing.
Conversely, A function f(x) decreases on an interval I if $f\left(b\right)\le f\left(a\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }b>a$, where $a,b\in I$. If $f\left(b\right)a$, the function is said to be strictly decreasing.
Then how would a horizontal line be described? If $f\left({x}_{2}\right)=f\left({x}_{1}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }{x}_{2}>{x}_{1}$ does that mean that a horizontal line meets the definition of an increasing function and a decreasing function? Reese King

Expert

Explanation:
Yes, an horizontal line is a function f such that $f\left(x\right)=c$ for all $x\in \mathbb{R}$, and, if $a\le b\to f\left(a\right)=c\le c=f\left(b\right)$. Idem for $\ge$. Nash Frank

Expert

Step 1
Inject $f\left(x\right)=c$ in the definitions:
"A function c increases on an interval I if $c\ge c\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }b>a$, where $a,b\in I$. If $c>c\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}b>a$, the function is said to be strictly increasing.
Step 2
Conversely, A function c decreases on an interval I if $c\le c\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }b>a$, where $a,b\in I$. If $ca$, the function is said to be strictly decreasing."
You should be able to see for yourself what property holds.

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