functions flashcards

Question 1

difference between codomain and image/range

Answer 1

codomain are the possible outputs

range/image are the actual outputs

Question 2

what is restriction

Answer 2

taking a smaller domain to only show part of the function

Question 3

what happens if you do composition with an identity function

Answer 3

nothing

Question 4

surjective

Answer 4

f(A) = B

if every image point has at least one preimage in A

Question 5

injective

Answer 5

if for all x,x’ЄA: x=x’ => f(x)!=f(x’) (all points have distinct images)
if every image point has at most one preimage point in A.

Question 6

finding the real valued inverse

Answer 6

(the function f: A-> B is injective)

1. consider the new function g: A->B where B = f(A), which has the same graph as f
2. g is bijective so it has an inverse g^-1: B->A
3. the real valued inverse is the function h: B-> ℝ which has the same graph as g^-1

Question 7

strictly increasing/ decreasing function are…

Answer 7

invertible

Question 8

Supremum (supA)

Answer 8

if A is bounded above, then the minimum of the upperbounds of A is called the supremum

Question 9

Infimum (infA)

Answer 9

If A is bounded below, then the maximum of the lowerbounds of A is called the infimum

Question 10

maximum (maxA)

Answer 10

largest defined element of A

Question 11

minimum (minA)

Answer 11

smallest defined element of A

Question 12

if A is unbounded above what is the supremum and maximum

Answer 12

supA = ∞

no maximum

Question 13

if A in unbounded below what is the infimum and minimum

Answer 13

infA = -∞

no minimum

Question 14

tending to infinity definition

Answer 14

Let f:A->ℝ, A⊆ℝ ( Assuming unbounded above)

we say that the lim x->∞ f(x) = ∞ if for all M>0 there exists ℕЄℝ such that for all xЄA if x>N then f(x)>M

Question 15

tending to - infinity definition

Answer 15

Let f:A->ℝ, A⊆ℝ ( Assuming unbounded above)

we say that the lim x->∞ f(x) = -∞ if for all M>0 there exists ℕЄℝ such that for all xЄA if x>N then f(x)>-M

Question 16

tending to a point definition

Answer 16

Let f:A->ℝ, A⊆ℝ ( Assuming unbounded above)
we say that the lim x->∞ f(x) = l (where l⊆ℝ) if for all Ɛ>0 there exists NЄℝ such that, for all xЄℝ if x>N then |f(x)-l| is less than epsilon

Question 17

function tends to infinity as x tends to a definition

Answer 17

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) we say that lim x->a of f(x)=∞ if for all M>0 there exists 𝛿>0 such that for all xЄA: if 0 is less than |x-a| is less than delta then f(x) is greater than M

Question 18

function tends to minus infinity as x tends to a definition

Answer 18

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) we say that lim x->a of f(x)=-∞ if for all M>0 there exists 𝛿>0 such that for all xЄA: if 0 is less than |x-a| is less than delta then f(x) is less than -M

Question 19

function tends to a limit l as x tends to a definition

Answer 19

let f:A->ℝ, A⊆ℝ, aЄℝ (assume that a is an acculmulation point of A) we say that lim x->a of f(x)=l if for all Ɛ>0 there exists 𝛿>0 such that for all xЄA: if 0 is less than |x-a| is less than delta then |f(x)-l| is less than epsilon

Question 20

continuity definition

Answer 20

let f:A->ℝ, A⊆ℝ.
Let aЄA. The function f is continuous at a if for all Ɛ>0 there exists 𝛿>0 such that for all xЄA: if |x-a| is less than delta then |f(x)-f(a)| is less than epsilon

Question 21

lim x->0 sinx/x =

Answer 21

= 1

Question 22

lim x->0 1-cosx/x^2 =

Answer 22

= 1/2

Question 23

lim x->∞ (1+1/x)^2 =

Answer 23

=e

Question 24

lim x-> -∞ (1+1/x)^2 =

Answer 24

=e

Question 25

lim x->0 log(1+x)/x =

Answer 25

=1

Question 26

lim x-> 0 e^x -1 / x =

Answer 26

=1

Question 27

lim x->∞ e^x/x^b

Answer 27

= ∞

Question 28

lim x->∞ logx/x^b =

Answer 28

= 0

Question 29

lim x->0+ x^blogx =

Answer 29

= 0

Question 30

A^B =

Answer 30

exp(BlogA)

Question 31

proving something tends to infinity

Answer 31

• write the definition
• use the definition backwards to find an expression for A/Ɛ
• ‘choose’ that value for A (in terms of Ɛ)
• then use definition forwards to show it tends to infinity

Question 32

Proving something is continuous

Answer 32

• write the definition
• use definition backwards to find an expression for 𝛿/Ɛ
• ‘choose’ that value for 𝛿 (In terms of Ɛ)
• then use definition forwards to show it is continuous

Question 33

proving something is differentiable

Answer 33

• to show that f is differentiable at x show that f’(x) exists at x

Question 34

f’ is convex =>

Answer 34

f’ is increasing

Question 35

f’ is concave =>

Answer 35

f’ is decreasing