Determine whether the following function is injective, surjective, or bijective.

Question

Determine whether the following function is injective, surjective, or bijective. For any bijection write down the inverse: f: R+ --> R+ given by f(x)=2^x

Explanation / Answer

(i) Injectivity. Suppose that there exist x, x' in X such that (g o f)(x) = (g o f)(x'). So, g(f(x)) = g(f(x')) ==> f(x) = f(x'), since g is injective ==> x = x', since f is injective. Since (g o f)(x) = (g o f)(x') ==> x = x', we conclude that (g o f) is injective. ----------- (ii) Surjectivity. Given (g o f): X --> Z, given any z in Z, we need to find x in X such that (g o f)(x) = z. g(f(x)) = z. The idea is to work inside out: Since g : Y --> Z is surjective, there exists y in Y such that g(y) = z. Now, given this y, we can use the surjectivity of f: X --> Y to assert that there exists x in X such that f(x) = y. We are done, because (g o f)(x) = g(f(x)) = g(y) = z, as required. ---------------- Bijectivity follows from both of the assertions (injectivity and surjectivity) above.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.