MAT 137 — LEC 0601 Instructor: Alessandro Malusà TA: Muhammad Mohid October 30th, 2020 Warm-up question : True or false? Suppose A and B are two sets, f : A → B and g : B → A functions. If, g ( f ( a )) = a for every a ∈ A , then f is invertible and f − 1 = g .
What can we actually conclude?
What can we actually conclude? 1 f is injective;
What can we actually conclude? 1 f is injective; 2 f is surjective;
What can we actually conclude? 1 f is injective; 2 f is surjective; 3 g is injective;
What can we actually conclude? 1 f is injective; 2 f is surjective; 3 g is injective; 4 g is surjective.
What is wrong with this proof? Theorem Suppose that I ⊆ R is an interval and f : I → I is an invertible and differentiable function, with f − 1 = f and f ′ ( x ) � = 0 for every x ∈ I . Then, for every x ∈ I , we have f ′ ( x ) = 1 or f ′ ( x ) = − 1.
What is wrong with this proof? Theorem Suppose that I ⊆ R is an interval and f : I → I is an invertible and differentiable function, with f − 1 = f and f ′ ( x ) � = 0 for every x ∈ I . Then, for every x ∈ I , we have f ′ ( x ) = 1 or f ′ ( x ) = − 1. Examples : I = R and f ( x ) = x , or f ( x ) = − x . Proof. Since f ′ ( x ) � = 0 for every x ∈ I , I can use the formula f − 1 � ′ = 1 f ′ = � f ′ . Now fix x ∈ I : using again that f ′ ( x ) � = 0, I can multiply both sides of the � 2 = 1, so f ′ ( x ) = ± 1. � f ′ ( x ) equation by f ′ ( x ) and find that
Worm up A worm is crawling accross the table. The path of the worm looks something like this: True or False? The position of the worm is a function.
Worm function A worm is crawling accross the table. For any time t , let f ( t ) be the position of the worm. This defines a function f . 1 What is the domain of f ? 2 What is the codomain of f ? 3 What is the range of f ?
Before next class... • Watch videos 4.5, 4.7, 4.8 and 4.9. • Download the next class’s slides (no need to look at them!)
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