MathCS.org - Real Analysis: Example 1.2.5(d):

Example 1.2.5(d):

Let f(x) = x² with domain and range being R. Find image and preimage of [-2, 2].

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If we draw the graph of the function and shade the interval [-2, 2] on the x axis (in red) then its projection onto the y axis will be the image (in green).

image([-2, 2]) = [0, 4]

If we draw the graph of the function and shade the interval [-2, 2] on the y axis (in red) then its projection onto the x axis will be the preimage (in green).

preimage([-2, 2]) = [-sqrt(2), sqrt(2)]

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Example 1.2.5(d):