Prove.(a) f : Z+ → Z+, f (x) = 2x + 1 is one-to-one.(b) f : Z+ → Z, f (x) = 2x − x2 is not one to one.(c) f : R \ {0} → R+, f (x) = 1x2 is onto.(d) f : Z+ → Z+, f (x) = 2x − 1 is not onto.

f: R+→R defined by f (x)=2x, x∈ (0,1), f (x) = 3x, x∈ [1,∞) is one -one, onto neither one-one nor onto  one-one, not onto onto

Consider the function f: R→R defined by f(x)=sin(x)+cos(2x). Which of the following statements about f(x) is true? a. The function f(x) is one-to-one (injective). b. The function f(x) is onto (surjective). c. The function f(x) is both one-to-one and onto. d. The function f(x) is neither one-to-one nor onto.

5. Determine whether each of these functions from Z to Z is one-to-one (onto)a) f (n) = n − 1 b) f (n) = n2 + 1 c) f (n) = n3 d) 2nf n     6. Determine whether f : Z × Z → Z is onto ifa) f (m, n) = 2m − n b) f (m, n) = m2 − n2 c) f (m, n) = m + n + 1d) f (m, n) = |m| − |n| e) f (m, n) = m2 − 4 f) f (m, n) = m + n7. Determine whether each of these functions is a bijection from R to R.a) f (x) = −3x + 4 b) f (x) = −3x2 + 7 c) f (x) = (x + 1)/(x + 2) d) f (x) = x5 + 18. Let S = {−1, 0, 2, 4, 7}. Find f (S) ifa) f (x) = 1 b) f (x) = 2x + 1 c) 5xf x     

Show that the function f:N→N, given by f(1)=f(2)=1 and f(x)=x−1 for every x>2, is onto but not one-one.Easy

Let X, Y and Z be any non-empty sets and let f and g be one-one functions of X onto Y andY onto Z respectively so that f and g are both invertible. Then, show that(a) g ◦ f is one-one(b) g ◦ f is onto(c) (g ◦ f )−1 = f −1 ◦g−1