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Draw the graph of |q(−x)| and of |3 − q(x + 2)|. com 28 Fig. 16 Fig. 17 Fig. 1 Injective, Surjective and Bijective Mappings Given the map f : A → B, and I ⊂ A, the set f (I ) = { f (x) : x ∈ I } is called the image of I under f . If I = A, then f (A) is called the image of f , or the range of f , and denoted Im( f ). Observe that f (A) ⊂ B but that, in general, f (A) = B. 1 The map f : A → B is called surjective if f (A) = B, that is, if for every b ∈ B there exists a ∈ A such that f (a) = b, and it is called injective if it never sends distinct points into the same point, that is, if f (a1 ) = f (a2 ) for any a1 , a2 ∈ A with a1 = a2 .

16 Consider the function f : (−∞, +∞) → (−π/2, π/2] defined by ⎧ π π ⎪ ⎪ ⎨arctanπx x ∈ (−∞, − 4 ] ∪ [ 4 , +∞) x ∈ (− π4 , 0] f (x) = 2x + ⎪ 2 ⎪ ⎩−2x x ∈ (0, π4 ). (a) Draw the graph of f . (b) Establish whether f is injective and/or surjective. (c) Establish in which intervals f is increasing. 17 On which intervals, if any, is the function f (x) = e−x log x − 5 injective? ex . e2x + ex + k (a) For which values of the real parameter k the function f is defined on R? (b) Put k = 1. Find a neighborhood of x0 = −1 in which f is invertible and write an explicit analytic expression of the inverse.

For any y0 ∈ R, f −1 ({y0 }) = {x ∈ I : (x, y0 ) ∈ Γ ( f )}. Therefore, the preimage of a point is found by considering the horizontal line y = y0 and then by collecting all the abscissae of the points that lie on the intersection between the horizontal line and Γ ( f ). It follows in particular that f is injective if and only if every horizontal line intersects Γ ( f ) in at most one point and it is surjective if and only if every horizontal line intersects Γ ( f ) in at least one point. Therefore, f is bijective if and only if every horizontal line intersects Γ ( f ) in exactly one point.

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Abelian integrals by George Kempf


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