Ch. 7 Functions, limits and continuity

Question 1

Let f:X –> Y be a function and Yo ⊂ Y. Define the preimage of the set Yo under f

Answer 1

The preimage is a subset of X and given by

f^(-1) (Yo) = {x ∈ X | f(x) ∈ Yo}

Question 2

What does injective mean?

Answer 2

Injective if f(x1) = f(x2)
–> x1 = x2
Every x value has its own unique corresponding y value, but not all y values must have a corresponding x

Question 3

What does surjective mean?

Answer 3

Surjective if every y value has at least one corresponding x value

Question 4

What does bijective mean?

Answer 4

Surjective and injective

Question 5

What functions have inverses?

Answer 5

Injective functions

Question 6

If f:[a,b] –> R is continuous, then…

Answer 6

(a) f is a bounded function

b) Sup(f) exists and there exists a c ∈ [a,b] with f(c) = sup(f

Question 7

f : R^3 --> R, f(x, y, z) = x^2 + y^2 + z^2
Find
(a) f^-1({1})
(b) f^-1({0})
(c) f^-1({1})
(d) f^-1([1,2])

Answer 7

(a) x^2 + y^2 + z^2 = -1
has no solution so answer is { }

(b) x^2 + y^2 + z^2 = 0
has only solution {(0,0,0)}

(c) x^2 + y^2 + z^2 = 1
has infinitely many solutions:
{(x,y,z) ∈ R | x^2 + y^2 + z^2 = 1}

(d) 1 ≤ x^2 + y^2 + z^2 ≤ 2
has infinitely many solutions:
{(x,y,z) ∈ R | } 1 ≤ x^2 + y^2 + z^2 ≤ 2}

Question 8

How to check if a function is continuous at c

Answer 8

If a function is continuous and f(c) = k
then f(xn) --> k for all sequences where xn --> c