MHB What is the difference in writing the range of values of a function

[mathlearn]

I was just wondering in writing the range of values for which the function is increasing or decreasing in in a positive or a negative way ,

The difference caused by the use of the symbols $'>'$ and $'≥'$ or in $'<'$ or $'≤ '$​

For example If we consider the graph of the function,

$y=\left(x-x\right)^2-5$ & asked to write down the interval of values of $x$ on which the function increases from $-5$ to $3$

The range can be written as & note here that $'≤'$ is used instead of $<$

$2≤ x≤ 4.8$

& taking another graph in the form of $y=2-x(x-4)-2$

write down the interval of $x$ in which the function is positive and increasing

$-1<x<1$

So what is the difference in the use of symbol $<$ & $≤$ in writing the range

 

[I like Serena]

Re: What is the difference in writing the range of values of a funtion

I was just wondering in writing the range of values for which the function is increasing or decreasing in in a positive or a negative way ,

The difference caused by the use of the symbols $'>'$ and $'≥'$ or in $'<'$ or $'≤ '$​

For example If we consider the graph of the function,

$y=\left(x-x\right)^2-5$ & asked to write down the interval of values of $x$ on which the function increases from $-5$ to $3$

The range can be written as & note here that $'≤'$ is used instead of $<$

$2≤ x≤ 4.8$

& taking another graph in the form of $y=2-x(x-4)-2$

write down the interval of $x$ in which the function is positive and increasing

$-1<x<1$

So what is the difference in the use of symbol $<$ & $≤$ in writing the range

Hey mathlearn! ;)

Both forms are fine and usually mean the same thing.
The difference is whether we want to include the end points or not.
And that difference is only relevant if the function is not defined or makes a jump at an end point.
If it's already known that the function is well defined and continuous, as in your examples, that can't happen.

 

[mathlearn]

Re: What is the difference in writing the range of values of a funtion

Thank you very much ILS :)

Hey mathlearn! ;)

Both forms are fine and usually mean the same thing.
The difference is whether we want to include the end points or not.
And that difference is only relevant if the function is not defined or makes a jump at an end point.
If it's already known that the function is well defined and continuous, as in your examples, that can't happen.

Hey I like Serena :D,

So then when we are asked to write an exact range like in,

$y=\left(x-x\right)^2-5$ & asked to write down the interval of values of $x$ on which the function increases from $-5$ to $3$

The range can be written as & note here that $'≤'$ is used instead of $<$

$2≤ x≤ 4.8$

we must be using the $≤ $ symbol

or if we are asked to write the range of values of which the function is increasing or decreasing negatively or positively in which we aren't given a limit using numbers we use $<$, Like in

write down the interval of $x$ in which the function is positive and increasing

$-1<x<1$

So what is the difference in the use of symbol $<$ & $≤$ in writing the range

Many Thanks (Smile)

 

[I like Serena]

Re: What is the difference in writing the range of values of a funtion

So then when we are asked to write an exact range like in,

$y=\left(x-x\right)^2-5$ & asked to write down the interval of values of $x$ on which the function increases from $-5$ to $3$

The range can be written as & note here that $'≤'$ is used instead of $<$

$2≤ x≤ 4.8$

we must be using the $≤ $ symbol

The text "from $-5$ to $3$" is ambiguous to whether the end points are included, so we are free to pick either. ;)

or if we are asked to write the range of values of which the function is increasing or decreasing negatively or positively in which we aren't given a limit using numbers we use $<$, Like in

write down the interval of $x$ in which the function is positive and increasing

$-1<x<1$

So what is the difference in the use of symbol $<$ & $≤$ in writing the range

Many Thanks (Smile)

In this case we should certainly not include -1 or 1 in the range, since they are specifically excluded. (Nerd)