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Introduction to Math Proof

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Question No. 4: Let f , g : R→R be functions defined by f(x)=x^3+x^2 and g(x)=4x-9. Find f∘g and g∘f. What can you conclude about the commutativity of the operation?

Question No. 5: Suppose f : X→Y and g : Y→Z are functions. If f and g are one-to-one, then so is g∘f.

Nov 19th, 2017

4.  (f∘g)(x) = f(g(x)) = [g(x)]^3 + [g(x)]^2 = [g(x)]^2*([g(x)]+1) = (4x-9)^2*(4x-8) = 4(x-2)(16x^2 - 72x + 81) =
4(16x^3 + 40x^2 + 225x -162).
g∘f(x) = g(f(x)) = 4f(x) - 9 = 4x^3 + 4x^2 - 9.
So the operation is NOT commutative.

5. Suppose "one-to-one" means "injective", U(t1) = U(t2) only if t1 = t2.
So let (g∘f)(x1) = (g∘f)(x2), g(f(x1)) = g(f(x2)).
g is one-to one implies that f(x1) = f(x2).
Now f is one-to-one gives that x1 = x2, which we ought to prove.

(if "one-to-one" means "bijective" than we use inverses for f and g, I'll write if you want)

Apr 27th, 2015

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