Functions

Question 1

range

Answer 1

is the set of all the images of f
subset of codomain
Ran f = { f(x) |x ∈ Dom f } c Codom f
codomain isnt mentioned assume its R
domain isnt mentioned assume it to be the maximal subset of R for which the map is defined

Question 2

function

Answer 2

f: D-> C is a map from a set D (domain) to a set C (codomain)
for any x in D there is a unique f(x) in C (called the image of x under f)
f: D -> C
x -> f(x)

Question 3

eg range, codomain, domain of x^2

Answer 3

f: R -> R is given by f(x) = x^2
so Dom f = R, Codom f = R and Ran f = [0, inf) c R

Question 4

the graph of a function

Answer 4

is the set of all points (x,y) ∈ R^2 (the xy plane) such that x is in the Dom f and y = f(x)
(we often draw it over an interval restricting the function to the same interval - often pick one to include important features)
is a curve but not every curve is the graph of a function

Question 5

included vs excluded points

Answer 5

shaded circle to indicate an included point
unshaded circle to denote excluded

Question 6

vertical line test

Answer 6

if any vertical line intersects curve more than once NOT A FUNCTION

Question 7

even function

Answer 7

f(x) = f(-x)
reflectionaly symmetric

Question 8

odd function

Answer 8

f(x) = -f(-x)
or f(-x) = -f(x)
rotationaly symmetric

Question 9

even and odd functions

Answer 9

all functions can be written as the sum of an even + odd function
f(x) = f even(x) + f odd(x)
f even(x) = 1/2(f(x) + f(-x))
f odd(x) = 1/2(f(x) - f(-x))

Question 10

piecewise functions

Answer 10

diff expressions for diff intervals of the domain
eg f(x) = |x|
x if x>= 0
-x if x <0

Question 11

step function

Answer 11

common piecewise function
constant on each piece
Eg Heaviside step
0 x<0
1 x>0

Question 12

operations with functions

Answer 12

sum
difference
scalar multiplication
linear combination
product
ratio
composition

Question 13

sum (f + g)

Answer 13

f+g(x) = f(x) + g(x)
Dom(f + g) = Dom f ⋂ Dom g

Question 14

Difference (f -g)

Answer 14

f -g(x) = f(x) - g(x)
Dom(f-g) = Dom f ⋂ Dom g

Question 15

scalar multiplication (cf)

Answer 15

(cf)(x) = c*f(x)
Dom(cf) = Dom f

Question 16

linear combinations (af + bg)

Answer 16

(af + bg) (x) = af(x) + bg(x)
Dom(af + bg) = Dom f ⋂ Dom g

Question 17

product (fg)

Answer 17

(fg)(x) = f(x)*g(x)
Dom(fg) = Dom f ⋂ Dom g

Question 18

ratio (f/g)

Answer 18

(f/g)(x) = f(x)/g(x)
Dom f/g = (Dom f ⋂ Dom g) \ {x|g(x) = 0}

Question 19

composition (f֯ºg)

Answer 19

(fºg)(x) = f(g(x))
Dom fºg = {x ∈ Dom g | g(x) ∈ Dom f}

Question 20

surjective

Answer 20

a function is surjective if Ran f = C
(if for all y within C there exists x within Dom f such that f(x) = y)

Question 21

injective

Answer 21

f : D-> C is injective if for all x1,x2 within D x1=/=x2 then f(x1) =/= f(x2)

Question 22

horizontal line test

Answer 22

if no horizontal line intersects the graph of f more than once then f is injective otherwise its not

Question 23

bijective

Answer 23

a function f: D -> C is bijective if its both injective + surjective

Question 24

inverse function

Answer 24

a bijective function f: D-> C has a unique inverse f^-1 such that fºf^-1(x) = x = f^-1ºf(x)
Ran f^-1 = Dom f
Dom f^-1 = Ran f ( = Codom f)
y = f(x) iff f^-1(y) = x