Set theory assignment questions 2 – 3

2.        (a) For the function f : R→R defined by

√

f(x) = esin(2 x+1)

find three functions g₁,g₂,g₃ : R→R so that f = g₁g₂g₃. Then find three different functions h₁,h₂,h₃ : R→R so that f = h₁h₂h₃. In both cases, none of your three functions should be the identity function (g(x) = x) and (as always) you should show all calculations necessary to verify your claims. (That is, after you give your functions, you should explicitly show that when they are composed together, they give f).

(b) Define sets A,B,C,D by

A = {a,b,c,d},B = {x,y,z},C = {p,q,r,s,t},D = {m,n}.

Let f : A → D be defined by

f(a) = m,f(b) = m,f(c) = n,f(d) = n.

Find functions g₁ : A → B, g₂ : B → C, and g₃ : C → D so that f = g₁g₂g₃.

3. Just because two functions have a certain property does not mean that their composite also has that property. For each of the following five properties of functions from R to R, your task is to determine whether or not the composite of two functions with that property also has that property.

For example, for part (a), you need to determine whether or not if f₁,f₂ : R→R are both increasing, then their composite f₁f₂ is also increasing. If you believe it is true, you should give a proof; if you believe it is false, you should give a specific counter-example.

(a) Increasing: functions f : R→R such that for any x₁ ≤ x₂, f(x₁) ≤ f(x₂).

(b) Decreasing: functions f : R→R such that for any x₁ ≤ x₂, f(x₁) ≥ f(x₂).

(c) Affine: functions f : R→R such that there exist constants a,b ∈R so that for any x, f(x) = ax + b. (For example, f₁(x) = 3x − 7 and f₂(x) = 5x are affine, but f(x) = x² − 8 is not).

(d) Quadratic: functions f : R→R such that there exist constants a,b,c ∈R ( with a 6= 0) so that for any x, f(x) = ax² + bx + c.

(e) Functions f : R→R such that f(0) ≤ 1.