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 ( b ) ≥ f ( a ) ∀ b &gt; a, where a , b ∈ I. If f ( b ) &gt; f ( a ) ∀ b &gt; a, the function is said to be strictly increasing.Conversely, A function f(x) decreases on an interval I if f ( b ) ≤ f ( a ) ∀ b &gt; a, where a , b ∈ I. If f ( b ) &lt; f ( a ) ∀ b &gt; a, the function is said to be strictly decreasing.Then how would a horizontal line be described? If f ( x 2 ) = f ( x 1 ) ∀ x 2 &gt; x 1 does that mean that a horizontal line meets the definition of an increasing function and a decreasing function?

Question

Is a horizontal line an increasing or decreasing function?This is the definition of an increasing and decreasing function.&quot;A function f(x) increases on an interval I if   f  (  b  )  ≥  f  (  a  )      ∀  b  &amp;gt;  a, where   a  ,  b  ∈  I. If   f  (  b  )  &amp;gt;  f  (  a  )      ∀      b  &amp;gt;  a, the function is said to be strictly increasing.Conversely, A function f(x) decreases on an interval I if   f  (  b  )  ≤  f  (  a  )      ∀  b  &amp;gt;  a, where   a  ,  b  ∈  I. If   f  (  b  )  &amp;lt;  f  (  a  )      ∀      b  &amp;gt;  a, the function is said to be strictly decreasing.Then how would a horizontal line be described? If   f  (      x    2    )  =  f  (      x    1    )      ∀      x    2    &amp;gt;      x    1   does that mean that a horizontal line meets the definition of an increasing function and a decreasing function?

Reese King · Accepted Answer

Explanation:Yes, an horizontal line is a function f such that   f  (  x  )  =  c for all   x  ∈      R  , and, if   a  ≤  b  →  f  (  a  )  =  c  ≤  c  =  f  (  b  ). Idem for   ≥.

Nash Frank · Accepted Answer

Step 1Inject   f  (  x  )  =  c in the definitions:&quot;A function c increases on an interval I if   c  ≥  c      ∀  b  &amp;gt;  a, where   a  ,  b  ∈  I. If   c  &amp;gt;  c      ∀      b  &amp;gt;  a, the function is said to be strictly increasing.Step 2Conversely, A function c decreases on an interval I if   c  ≤  c      ∀  b  &amp;gt;  a, where   a  ,  b  ∈  I. If   c  &amp;lt;  c      ∀      b  &amp;gt;  a, the function is said to be strictly decreasing.&quot;You should be able to see for yourself what property holds.