9.4 Problems

Problem 1. Which of these functions are injective, surjective, or bijective? Find the inverse of bijective functions.

(a) f : ℝ → (0,∞), x→e^x (b) g : ℝ → [0,∞), x→x² (c) h : [0,∞) → [0,∞), x→x² (d) sin : ℝ → [0,1], x→sin(x) (e) tan : ℝ →ℝ, x→tan(x),

Problem 2. Find a function f : ℝ →ℝ such that (f ∘f)(x) = −x.

Problem 3. Can any real function g : ℝ →ℝ be obtained as g = f ∘f for some f : ℝ →ℝ?

Problem 4. Following the example of the composition Section 9.2.1, implement

• the add function, taking f and g, returning f + g,
• the mul function, taking f and g, returning fg,
• and the div function, taking f and g, returning f∕g.

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