solve 1. (15 pts.) Note the contrapositive of the definition of one-to-one funct
ID: 2962802 • Letter: S
Question
solve
1. (15 pts.) Note the contrapositive of the definition of one-to-one function given on page 141 of the text is: If a b then f(a) f(b). As we know, the contrapositive is equivalent to (another way of saying) the definition of one-to-one (a) Consider the following function f R R defined by fx) x 9. Use the contrapositive of the definition of one-to-one function to determine (no proof necessary) whether f is a one-to-one function. Explain (b) Compute f o (c) Let g be the function g: R R defined by go) x 3. Find g 1. Use the definition of g to explain why your solution, g is really the inverse of g.Explanation / Answer
(a)
2 != -2
but
f(-2) =-5 = f(2)
=.
f is not one-to-one
(b)
fof = (x^2-9)^2-9 = x^4-18x^2+72
(c)
g^(-1) (x) = (x-3)^(1/3)
by definition, go(g^(-1))(x)) = x
lets check it:
go(g^(-1))(x)) =g[(x-3)^1/3] = [(x-3)^1/3]^3 +3 = (x-3)+3 = x
thus proved
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