# How to map intervals of Real line

• Diffy
In summary, creating a function that maps an interval of the real line onto another interval can be done using a linear map in most cases. However, for infinite or half-infinite intervals, other functions such as exp() or tan() may need to be used.
Diffy
Can someone explain how to create a function that will map an interval of the real line onto some other interval?

Is there a general method?

Can you demonstrate? (30 140) to (200, 260)?

Thanks,

Diffy.

try with a linear map.

Let x1 be in (30, 140) and let x2 be in (200, 260)
then f: x1 -> x2
f(x) = (6/11)(x-30) + 200

Something like that?

Will this always work for (x1, x2) -> (y1, y2)?

f(x) = (y2 - y1)/(x2 - x1) * (x - x1) + y1

Indeed, let us prove it as follws:

1. For any X so that x1<=X<=x2, we have y1<=f(X)<=y2
Proof:
We clearly have :
y1=(y2-y1)/(x2-x1)*(x1-x1)+y1<=f(X)<=(y2-y1)/(x2-x1)*(x2-x1)+y1=y2

2. Similarly, we can show that for any Y so that y1<=Y<=y2, then there exists some X, so that x1<=X<=x2 AND that f(X)=Y.

We get that X=(Y-y1)/(y2-y1)*(x2-x1)+x1. Since 0<(Y-y1)/(y2-y1)<1, that result follows immediately.

Thank you.

In most cases, you can use a linear map. Some trickiness results if one of the intervals is infinite or half-infinite. In that case, you might have to use exp(), tan(), or their inverses.

## 1. What is the Real line?

The Real line, also known as the number line, is an infinitely long line that contains all the possible real numbers, including positive and negative numbers, as well as zero. It is a fundamental concept in mathematics and is used to represent quantities and measurements.

## 2. How do you map intervals on the Real line?

To map intervals on the Real line, you need to first identify the starting and ending points of the interval. Then, you can mark these points on the number line and draw a line connecting them. The space between these two points represents the interval. The endpoints can be included or excluded, depending on the type of interval.

## 3. What is the difference between open and closed intervals?

An open interval does not include its endpoints, while a closed interval includes its endpoints. For example, the interval (0, 5) would include all real numbers greater than 0 and less than 5, but not including 0 and 5. On the other hand, the interval [0, 5] would include all real numbers greater than or equal to 0 and less than or equal to 5, including 0 and 5.

## 4. How do you represent infinite intervals on the Real line?

Infinite intervals can be represented by using the symbols ∞ and -∞. For example, the interval (-∞, 5) would include all real numbers less than 5, while the interval (3, ∞) would include all real numbers greater than 3.

## 5. Why is it important to understand how to map intervals on the Real line?

Mapping intervals on the Real line is important in various fields of science, such as physics, economics, and statistics. It allows us to visualize and represent data and make accurate calculations. It also helps in solving problems and making predictions based on real-world scenarios.

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