Monotonically decreasing/increasing

[BOAS]

Homework Statement

Plot each function and decide, based on the plot whether or not it is monotonically increasing/decreasing or strictly monotonically increasing/decreasing.

f(x) = -5

Homework Equations

The Attempt at a Solution

I can plot this function no problem, and show algebraically why it fits the definition of monotonically increasing and monotonically decreasing.

My question is, how do I justify this 'based on the plot'?

The next part of the question is to show it algebraically, but graphically I'm not really sure why my answer is correct...

Thanks for any help you can give.

 

[Ray Vickson]

You really need to show more: what exact definitions are you using for "monotonically increasing/decreasing"? You seem to be making a distinction between monotonically increasing and strictly monotonically increasing, etc. Your answer could be right or wrong, depending on these details.

 

[Mark44]

Homework Statement

Plot each function and decide, based on the plot whether or not it is monotonically increasing/decreasing or strictly monotonically increasing/decreasing.

f(x) = -5

Homework Equations

The Attempt at a Solution

I can plot this function no problem, and show algebraically why it fits the definition of monotonically increasing and monotonically decreasing.

My question is, how do I justify this 'based on the plot'?

The next part of the question is to show it algebraically, but graphically I'm not really sure why my answer is correct...

The question distinguishes between monotonically increasing/decreasing and strictly monotonically increasing/decreasing. In the latter, the graph of the function would have to be trending up (mon. increasing) or down (mon. decreasing). From your plot, which should show a horizontal line, which you could characterize as both monotically decreasing and monotonically increasing.

 

[BOAS]

You really need to show more: what exact definitions are you using for "monotonically increasing/decreasing"? You seem to be making a distinction between monotonically increasing and strictly monotonically increasing, etc. Your answer could be right or wrong, depending on these details.

I have attached the definitions I am using as a picture. I will explain how I'm using them to come to my result.

From this, I say that the function $f(x) = -5$ is monotonically increasing and monotonically decreasing because for all $x_{1}, x_{2} \in \mathbb{R}$ where $x_{1} < x_{2}, f(x_{1}) \leq f(x_{2})$.

Likewise, for all $x_{1}, x_{2} \in \mathbb{R}$ where $x_{1} < x_{2}, f(x_{1}) \geq f(x_{2})$.

 

[Mark44]

I figured that these were the definitions you were using. I agree with your answer.

 

[vela]

The next part of the question is to show it algebraically, but graphically I'm not really sure why my answer is correct.

Is your question essentially "How can a flat graph be considered increasing or decreasing?"

 

[BOAS]

that is at the heart of it.

 

[vela]

It's just how it is. As long as the function satisfies the definition, we say it's monotonically increasing or decreasing, even if it may run counter to our everyday sense of the words. Your intuition aligns with strictly monotonically increasing or decreasing.

 

[Mark44]

that is at the heart of it.

Note that the definitions for monotonically increasing and decreasing include the possibility that f(x₁) = f(x₂). IOW, they include the possiblity of graphs that are flat. The "strictly" definitions don't include this.