Let p = 2k +1 be an odd prime, S = {0, 1, 2, ..., k}, and Z, = {0,1,2,.,p – 1}. Define g : S → Z, by g(y) = (y² + 1)%p. Let p= 11. Draw arrow diagrams for g and for f :S→ Zp given by f(x) = (-r²)%p. Use your diagrams to find all ordered pairs (r, y) with r, y E S and r+ y² +1 =0 (mod 11).

6. Let f(x)=x55x+1. Prove that f has at least 3 zeros in [-2, 2]; that is, there exist distinct ₁, 2, 3 € [-2,2] such that f(x₁) = f(x₂) = f(x3) = 0.

Z is the set of integers, R is the set of real mumbers. 1. Functions f, 9, and h are defined as f:R + R, g :R + R, h:R - R, h(r) = Vr? +1 f(1) = 1+3 g(r) = +1 %3D Find: (a) (ho f)(2) (b) (fo f)(z) (c) (go f)(x)

5. (a). u. (7v+w); (b). (uw)w||; (c). ||u|| (vw); (d). (uv). w. Let & (1, 2, 0), 7(4, 1, -2), and w= (0, 2,3). Find

Find the general formula for T: R3 -> R3, if T(1,0,1) = (1,2,0) , T(1,-2,1) = (0,1,0) and T (0,0,1) = (0,2, -1). Also find T(2,-3,1).

. (블블) k – 1 k+1 Let h : Z → Q × Q, given by h(k) : 2 2 Let G = {(m, m + 1) | m E Z}. Show that G C h(Z).

Determine a and b such that p(x) = x² + ax + b satisfies p(1) = 0 and p'(1) = 9. (Give an exact answer. Use symbolic notation and fractions where needed.) a = b =

1. Let A={1,3,5,7}. Let f:A→Z be the function f={(1,2),(3,4),(5,6),(7,8)}. Let g:Z→Z be defined by g(n) = 2n-1. Find gof.

4. (a) Let a Z→→ Z be the function defined by a(n) = 2n + 1, and let i: Z→ Z be the identity function. Is there a function b: Z→ Z such that bo a = i? Is there a function b: Z→ Z such that a ob=i? (b) Let f: N N and g: N→ N be defined by f(n) = n + 1, n-1 g(n) = 1 Find the functions fog, gof, fogof and gofog. if n > 1 if n = 1.

Let C = {0, 1, 2, 3} and f: C→C, g: C→C with f = {(0, 2), (1, 2), (2, 3), (3, 0)}, 9 = {(0, 3), (1, 1), (2, 0), (3, 2)}. (a) Write down the compositions fog, gof and gog.