functions flashcards
difference between codomain and image/range
codomain are the possible outputs
range/image are the actual outputs
what is restriction
taking a smaller domain to only show part of the function
what happens if you do composition with an identity function
nothing
surjective
f(A) = B
if every image point has at least one preimage in A
injective
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.
finding the real valued inverse
(the function f: A-> B is injective)
- consider the new function g: A->B where B = f(A), which has the same graph as f
- g is bijective so it has an inverse g^-1: B->A
- the real valued inverse is the function h: B-> ℝ which has the same graph as g^-1
strictly increasing/ decreasing function are…
invertible
Supremum (supA)
if A is bounded above, then the minimum of the upperbounds of A is called the supremum
Infimum (infA)
If A is bounded below, then the maximum of the lowerbounds of A is called the infimum
maximum (maxA)
largest defined element of A
minimum (minA)
smallest defined element of A
if A is unbounded above what is the supremum and maximum
supA = ∞
no maximum
if A in unbounded below what is the infimum and minimum
infA = -∞
no minimum
tending to infinity definition
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
tending to - infinity definition
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
tending to a point definition
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
function tends to infinity as x tends to a definition
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
function tends to minus infinity as x tends to a definition
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
function tends to a limit l as x tends to a definition
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
continuity definition
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
lim x->0 sinx/x =
= 1
lim x->0 1-cosx/x^2 =
= 1/2
lim x->∞ (1+1/x)^2 =
=e
lim x-> -∞ (1+1/x)^2 =
=e
lim x->0 log(1+x)/x =
=1
lim x-> 0 e^x -1 / x =
=1
lim x->∞ e^x/x^b
= ∞
lim x->∞ logx/x^b =
= 0
lim x->0+ x^blogx =
= 0
A^B =
exp(BlogA)
proving something tends to infinity
- 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
Proving something is continuous
- 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
proving something is differentiable
- to show that f is differentiable at x show that f’(x) exists at x
f’ is convex =>
f’ is increasing
f’ is concave =>
f’ is decreasing
