Consider the function f: epsilon rightarrow epsilon obtained by f(x, y) = (-y, x

Question

Consider the function f: epsilon rightarrow epsilon obtained by f(x, y) = (-y, x^3 + y^2) for all (x, y) elementof epsilon. (a) Prove that f is a transformation and find a formula for f^-1(x, y) (b) Show that f is not isometry (c) Find all fixed points of f. (d) Describe the images of straight lines under the transformation f. Find the equations of the lines which one mapped either to a line or parabola (e). Find the equations of f(l) and f(m), where L is the line with equation y = 5 and m is the line with equation x = 0

Explanation / Answer

Given that :

f(x, y) = (-y, x^3 + y^2) for all x, y belongs to E

Then

(a) So f^-1 (x, y) =

let us say u = -y, v = x^3 + y^2

y = -u, x = (v-u^2)^(1/3)

f^-1(x,y) = (-x, (y-x^2)(1/3)) Ans

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