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Chapter 1: Problem 8

Let \(f(x):=1 / x^{2}, x \neq 0, x \in \mathbb{R}\) (a) Determine the direct image \(f(E)\) where \(E:=\\{x \in \mathbb{R}: 1 \leq x \leq 2\\}\). (b) Determine the inverse image \(f^{-1}(G)\) where \(G:=\\{x \in \mathbb{R}: 1 \leq x \leq 4\\}\).

Short Answer

Expert verified
The direct image of the function is [1/4, 1], and the inverse image is [-1, -0.5] together with [0.5, 1]

Step by step solution

01

Determine the direct image \(f(E)\)

To determine the direct image, plug the interval [1, 2] into the function \(f(x) = 1/x^2\). When \(x = 1\), \(f(1) = 1/1^2 = 1\). When \(x = 2\), \(f(2) = 1/2^2 = 1/4\). The result is a new interval of [1/4, 1]
02

Determine the inverse image \(f^{-1}(G)\)

The inverse image requires determining which \(x\) values satisfy \(1 <= 1/x^2 <= 4\). Solving this inequality reveals an interval for \(x\) in the range of [-1, -0.5] together with [0.5, 1]. The inverse image thus consists of these two intervals.

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