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Correction: all numbers between -2 and 2 (inclusive).HallsofIvy said:The given function was [itex]f(x)= x^2[/itex] and [itex]E= \{0\le x\le 2\}[/itex].
They got [itex]0\le f(x)\le 4[/itex] by squaring 0 and 2.
But be careful, if it had been [itex]E= \{-2\le x\le 2\}[/itex] We would NOT have the image [itex]\{4\le f(x)\le 4}[/itex]. [itex]-2\le x\le 2[/itex] includes all numbers from 0 to 2
HallsofIvy said:and the squares of those are all between 0 and 2, also the squares of the negatives are also positive, not negative: If [itex]E= \{-2\le x\le 2\}[/itex] we would still have the image as [itex]\{0\le f(x)\le 4\}[/itex]
A direct image in real analysis is a mathematical concept that involves mapping elements from one set to another set. It is also known as the image of a set, and is denoted as f(A), where f is a function and A is the set being mapped. The direct image is the resulting set of applying the function f to all the elements in set A.
An inverse image in real analysis is the preimage of a set. It involves mapping elements from the codomain of a function to the domain of the function. It is denoted as f-1(B), where f is a function and B is the set being mapped. The inverse image is the set of all elements in the domain that map to the set B in the codomain.
The direct and inverse images are related in that they are the opposite mappings of each other. The direct image maps elements from the domain to the codomain, while the inverse image maps elements from the codomain to the domain. They are both important concepts in real analysis and are used to study the behavior of functions.
The main difference between direct and inverse images is the direction of the mapping. In direct images, elements are mapped from the domain to the codomain, while in inverse images, elements are mapped from the codomain to the domain. Additionally, direct images are denoted as f(A), while inverse images are denoted as f-1(B).
Direct and inverse images are used in real analysis to study the behavior of functions and to prove theorems. They are also used to define important concepts such as continuity, compactness, and connectedness. In addition, they are used to define the fundamental concepts of open and closed sets in topology.