Exercises
- Give examples of "one-to-one" and "onto" functions.
- Verify whether the following functions are "one-to-one" or "onto"
mappings. Determine their domain and codomain of them.
- f(x) = ln|x| for x ∈ (-1,- ∞
) ∪ (1, + ∞
) and f(x)= x2 -1 for x ∈ [-1,1]
- f(x) = sqrt(x+1) for x ∈
R+ and f(x) = 2x for x ∈ R\R+
- f : R × R →
R, such that f((x,y)) = x + y
- f(x) = x/(x+1) for x ≠ -1 and f(-1)
= 1
- f(x) = x4 - 5x2 + 4 for x ∈
R
- Find the functions, the composition of which is f(x) = 3 (x2
+5)2 for x ∈ R?
- Find the inverse of the following functions:
- f(x) = 3x-2, for x ∈ R
- g(x) = ax2 +b, where x ∈
R+ and a, b are constants.
- h(x) = 1/(x+1) where x ∈ R and x ≠ 1
- Let f = (2 1 4 3 ) and g = (1 2 4 3 ) be two permutations of the
set {1,2,3,4}. Find ( f o g), ( f o f) and (g o f).
- Let f(x) = x2 -3x + 2, x ∈
R. Find f([-2,-1]), f({1,2}) and f-1([-1,0]), f-1([-3,-4]).
- Prove that the composition of two "one-to-one" functions is a
"one-to-one" function.
- Prove that for arbitrary function f and arbitrary sets A, B, f(A ∩ B) ⊆ f(A) ∩ f(B).
Give two examples of the diagrams of functions and mark on them the
sets f((A ∩ B), f(A), f(B), f(A) ∩ f(B). How the properties change if f is
a "one-to-one" function?
- Let f be a function from X to Y. Prove that f-1(A ∪ B) = f-1(A) ∪
f-1(B) for arbitrary subsets A, B of Y.
- Write a program that prints "YES" in the case when the function,
given
in the form of a table of values, is "one-to-one", and "NO" in the
opposite case.