CG

Question 1

What is the definition of a function?

Answer 1

A function f from X to Y is a relation on X and Y s.t for each x ∈ X there is exactly one y ∈ Y with (x,y) ∈ f, that is there for each x in X, there is exactly one y in Y s.t f(x) = y.

Question 2

With y = f(x), what is y?

Answer 2

The image of x under f and the dependant variable

Question 3

With y = f(x), what is x?

Answer 3

The independent variable

Question 4

(Let f : X->Y)

What is the set X called?

Answer 4

The domain of f, dom(f)

Question 5

(Let f : X->Y)

What is the set Y called?

Answer 5

The codomain of f, codom(f)

Question 6

(Let f : X->Y)

How would you write the set for im(f)?

Answer 6

im(f) = {y∈Y : y=f(x) for some x∈X}

Question 7

Another word for image?

Answer 7

Range, ran(f)

Question 8

In terms of the set of values, describe the domain, codomain and image of f.

Answer 8

Domain of f = set of input values for f
Codomain of f = set of possible outputs of f
Image of f = set of values that are actually output

Question 9

When is a rule well-defined or unambiguous?

Answer 9

Provided f(x) = f(x’) whenever x = x’

Question 10

When are two functions (f and g) equal?

Answer 10

1. dom(f) = dom(g)
2. codom(f) = codom(g)
3. ∀ x∈X, f(x) = g(x)

Question 11

Let f : X->Y and g : Y->Z be functions. What is the composition of f and g?

Answer 11

The function g∘f : X->Z defined by the rule g∘f(x) = g(f(x)).

Question 12

When is f the identity function on X?

Answer 12

If f :X->X and f(x) = x ∀x

Question 13

When is f called the constant function y?

Answer 13

If f :X->Y and f(x) = y ∀x∈X

Question 14

Let f: X->Y be a function. What is the inverse of f?

Answer 14

If it exists, f^-1 : Y->X with the property that

f(x) = y iff f^-1(y) = x.

Question 15

Let f: X->Y be a function. When is f surjective (onto)?

Answer 15

iff for every y∈Y, there is at least one x∈X s.t f(x) = y.

Question 16

Let f: X->Y be a function. When is f injective (one-to-one)?

Answer 16

iff for every y∈Y, there is at most one x∈X s.t f(x) = y.

Question 17

Let f: X->Y be a function. When is f bijective?

Answer 17

iff f is both surjective and injective

Question 18

Let f: X->Y be an injective function, if x ≠ x’, then f(x)?

Answer 18

≠f(x’)

Question 19

Let f: X->Y be an injective function, if f(x) = f(x’), then x?

Answer 19

=x’

Question 20

Let f: X->Y be a function. When is f onto? (relation between im and codom)

Answer 20

iff im(f) = codom(f)

Question 21

Let f: X->Y be a function. How do you show that f is a bijection?

Answer 21

We must show that it is both an injection and a surjection.

Question 22

Let f: X->Y be a function. What is the only condition for f to have an inverse?

Answer 22

f has an inverse iff it is a bijection

Question 23

If f is a bijection, what is the identity function on Y?

Answer 23

f∘f^-1

Question 24

If f is a bijection, what is the identity function on X?

Answer 24

f^-1∘f

Question 25

If X⊆|R then f is?

Answer 25

A function of a real variable

Question 26

If Y⊆|R then f is?

Answer 26

A real-valued function

Question 27

If X⊆|R and Y⊆|R then f is?

Answer 27

A real-valued function of a real variable

Question 28

Let f : |R->|R, f is a function iff

Answer 28

Every vertical line crosses the graph of f exactly once

Question 29

Let f : |R->|R, f is one-to-one iff

Answer 29

Every horizontal line crosses the graph of f at most once

Question 30

Let f : |R->|R, f is onto iff

Answer 30

Every horizontal line crosses the graph of f at least once

Question 31

Let f : |R->|R, f is a bijection iff

Answer 31

Every horizontal line crosses the graph of f exactly once

Question 32

What are the rules that both define real-valued functions of a real variable?

Answer 32

(a) h(x) = λf(x) + μg(x) and

b) k(x) = f(x)g(x

Question 33

What is the “Domain Convention”?

Answer 33

When only the rule of a real-valued function of a real variable f is defined. We assume that codom(f) is |R, dom(f) is the largest set of |R, for which the rule is valid.

Question 34

Let p>0. When is a real-valued function f : R->R of a real variable said to be periodic with period p?

Answer 34

If, for each x∈|R, f(x+p) = f(x).

Question 35

What is the fundamental period of f?

Answer 35

If a function f is periodic, the smallest positive p s.t f(x+p) = f(x) ∀ x∈X

Question 36

Let f : |R->|R be a real-valued function of a real variable.
When is f even and when is f odd?

Answer 36

f is said to be even iff f(-x) = f(x) for all x∈R.

f is said to be odd iff f(-x) = -f(x) for all x∈R.

Question 37

Suppose that f : X->Y be a function that is not a bijection. What are the steps to define a new function that does have an inverse?

Answer 37

Step 1: Restrict the co-domain of f to im(f). The resulting function is onto.
Step 2: Restrict the domain of f so that the resulting function is one-to-one.
Step 3: The resulting function, which, in general, will not be equal to f, is a bijection and therefore has an inverse.

Question 38

What are the domains and co-domains of the following functions?

1. f(x) = √x
2. f(x) = sin^-1(x)
3. f(x) = cos^-1(x)
4. f(x) = tan^-1(x)

Answer 38

1. |R0+, |R0+
2. [-1,1] , [-π/2,π/2]
3. [-1,1] , [0,π]
4. |R, (-π/2,π/2)

Question 39

Let f be a real-valued function of a real variable. 
Define f(x)->inf as x->inf

Answer 39

Suppose that f is defined ∀ x>r, for some r.
We say that the limit of f, as x tends to inf, is inf iff
∀ K>0, ∃ X s.t f(x)>K whenever x>X.

Question 40

Let f be a real-valued function of a real variable. 
Define f(x)->(-)inf as x->inf

Answer 40

Suppose that f is defined ∀ x>r, for some r.
We say that the limit of f, as x tends to inf, is (-)inf iff
∀ K<0, ∃ X s.t f(x)X.

Question 41

Let f be a real-valued function of a real variable. 
Define f(x)->inf as x->(-)inf

Answer 41

Suppose that f is defined ∀ x0, ∃ X s.t f(x)>K whenever x

Question 42

Let f be a real-valued function of a real variable. 
Define f(x)->(-)inf as x->(-)inf

Answer 42

Suppose that f is defined ∀ x

Question 43

Definition of a limit l, as x->inf,f(x)->l

Answer 43

Let f be a real-valued function of a real variable. Let l be a real number. Suppose f is defined ∀ x>r, for some r.
We say that the limit of f, as x tends to inf, is l iff
∀ε>0, ∃ X s.t | f(x)-l |X.

Question 44

Definition of a limit l, as x->(-)inf,f(x)->l

Answer 44

Let f be a real-valued function of a real variable. Let l be a real number. Suppose f is defined ∀ x0, ∃ X s.t | f(x)-l |

Question 45

Define f(x)->l as x->a

Answer 45

Let f be a real-valued function of a real variable. Suppose that b<a>0, there is a δ>0 s.t | f(x)-l | < ε ∃ whenever 0</a>

Question 46

Define the limit, f(x)->l as x->a_ (One-sided limit)

Answer 46

We say that the limit of f, as x tends to a from below or the left, is l iff
for every ε>0, there is a δ>0 s.t | f(x)-l | < ε ∃ whenever 0x

Question 47

Define the limit, f(x)->l as x->a+

Answer 47

We say that the limit of f, as x tends to a from above or the right, is l iff
for every ε>0, there is a δ>0 s.t | f(x)-l | < ε ∃ whenever 0a

Question 48

f(x)->inf as x->a

Answer 48

Suppose that b<a>0, there is a δ>0 s.t K</a>

Question 49

What is the squeeze rule?

Answer 49

Suppose that b<a>a = lim[h(x)] as x->a = l, then the limit of f as x->a exists and is equal to l.</a>

Question 50

Prove the squeeze rule

Answer 50

Let ε>0. Pick δ>0 s.t both | g(x)-l |

Question 51

Why does lim of sinθ/θ = 1 as θ->0?

Answer 51

Draw out circle, r=1. Then draw triangle,arc and bigger triangle, the areas of which are A,B,C respectivel y.
A≤B≤C obviously.
sinθ/2≤θ/2≤sinθ/2cosθ or 1≤θ/sinθ≤1/cosθ so that cosθ≤sinθ/θ≤1.
Since cosθ->1 as θ->0.

Question 52

What is the quotient rule?

Answer 52

d/dx (u/v) =(vu’-uv’)/v^2

Question 53

What’s d/dx of tanx?

Answer 53

sec^2(x)

Question 54

Define what it means for f to be continuous on (b,c)

Answer 54

We say that f is said to be continuous at a∈(b,c) iff for all ε>0, there exists δ>0 s.t |f(x)-f(a)|

Question 55

Suppose that f is differentiable on (b,c). Prove that f is continuous on (b,c).

Answer 55

Let a∈(b,c). We want to show f is cts at a, we need to show that lim(x->a)f(x)=f(a), which is he same as lim(x->a)(f(x)-f(a))=0
Since f is differentiable at a, we know that lim(h->0)(f(a+h)-f(a))/h exist. Call this limit l. Writing h=x-a, we have lim(h->0)(f(x)-f(a))/x-a = l.
Now f(x)-f(a) = f(x)-f(a) x (x-a)/(x-a)
Hence by AOL, lim(x->a)f(x)-f(a) = lim(x->a) f(x)-f(a) x (x-a)/(x-a)
= lim(x->a) f(x)-f(a)/(x-a) lim(x->a)x-a = l*0 =0 as required.