a) Prove algebraically that f: R → R, defined by f(x) = 3x is a 1-1, onto (bijective) map. Note that we have to prove that i) f(x1) = f (x2) → x1 = x2 b) For each y ER, we can find x E R such that f(x) = y

If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

b) Find Af(x) if f(x) = (3x+1)(3x+ 4)(3x+7)... (3x+ 19). / c) Let A = {1,2,3}, B = {a, b, c}, and C = {x, y, z}. Consider the following relations R and S from A to B and from B to C, respectively. R = {(1,b), (2,a), (2, c)} and S = {(a, y), (b,x), (c, y), (c, z) }. %3D %3D Find the matrix representation of S°R

Select the following functions that are bijective. f3: R → R, f(x) = 14 f₂: R → R, f₂(x) 2 f₁:R → R, f₁(x) = x + x + x + x + 4 f4: R → R, f4(x) = -{ = x x², if x 20 X, if x <0

Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X → V is injective. ) gof: X → V is surjective. ) (gof)-¹=f-¹og=¹ Use: v injective if x1 ‡xn → v(x1) = v(xn)

a) Give an example of an algebraic function f : Z → Z that is injective but not surjective. You do not need to defend your answer b) Let A = {a, b} and B = {1,2}. Give an element of A x B × A.

f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X → V is injective. ) gof: X→ V is surjective. ) (gof)-¹ = f¹og-¹ v injective if x1 #xn → v(x1) = v(xn)

a function. tion. (b) Let R= {(x,y) E R × R : 1 < x < 4 or a = 6}, and S = {(x,y) ER × R : –1 < y < 1}. i. Find the relations S1 and R-1 o S. ii. Represent the relations R, S, RUS and RnS in the (r, y)- plane.