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

AI Thread Summary
The discussion centers on the use of symbols like '<' and '≤' when writing the range of values for a function. The key difference lies in whether endpoints are included; '≤' indicates inclusion while '<' indicates exclusion. For continuous functions, as in the examples provided, the choice of symbol depends on the context of the problem. When specifying exact intervals, such as from -5 to 3, '≤' is appropriate if endpoints are included, while '<' is used for open intervals where endpoints are excluded. Understanding this distinction is crucial for accurately representing the behavior of functions.
mathlearn
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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
 
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Re: What is the difference in writing the range of values of a funtion

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

Thank you very much ILS :)

I like Serena said:
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,

mathlearn said:
$y=\left(x-x\right)^2-5$ & 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

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

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

mathlearn said:
$y=\left(x-x\right)^2-5$ & 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

mathlearn said:
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)
 
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