functions

Question 1

function

Answer 1

a rule f that assigns every x in a domain D, exactly one element f(x) in a codomain (set) E

arise whenever one quantity depends on another.

Question 2

test if something is a function

Answer 2

vertical line test
function can only have one output y, for each unique input x

Question 3

check if function is one to one

Answer 3

Horizontal line test : if any horizontal line intersects the graph only once then the function is one-to-one
f(x) = f(y) , x=y OR
x ≠ y , f(x) ≠ f(y)

Question 4

composite functions

Answer 4

if f & g are functions and the range of g(inner) overlaps with the domain of f, then we can compose these 2 functions

Question 5

even function

Answer 5

function is even if f(x) = f(-x) for every value of x in domain.

symmetric/reflection abt y axis

Question 6

odd function

Answer 6

a function is odd if -f(x) = f(-x) (OR f(x) = -f(-x) , same thing) for every value of x in the domain

symmetric about origin
rotation 180

Question 7

difference between range and codomain

Answer 7

codomain -all possible theoretical output values
range contained inside codomain

Question 8

when can composition take place

Answer 8

you are able to compose 2 functions g(f(a)) only if the range of the input is contained in the domain of output

Question 9

rational function

Answer 9

f(x) = P(x) / Q(x)
where Q(x) cannot = 0
degree of P(x) should be less than degree of Q(x)

Question 10

piece wise functions

Answer 10

function who’s domain is partitioned into several intervals on which the function may be defined differently

Question 11

absolute value function

Answer 11

type of piecewise function
f(input) = |input| = { input of input >=0, -(input) of input<0

Question 12

vertical asymptote

Answer 12

A vertical asymptote occurs at x=a if the graph moves infinitely up or down as x gets closer to a

Question 13

find domain of composite functions

Answer 13

Identify domain restrictions of the inner function
Find composite function
Identify domain restrictions of the composite function
Final domain includes both restrictions

OR (both do same thing)
Find domain of inner and outer
Domain = inner + value of x that would give domain of outer

Question 14

|5x +9| = -3

Answer 14

no real solution
cant get a negative value out of absolute value

Question 15

|2x-4| < 10

Answer 15

put pos and neg values on either side of eqn

-10 <2x-4< 10

Question 16

V asymptotes rational functions

Answer 16

set the denominator to zero and solve.

Question 17

inverse functions defn

Answer 17

reflection about y=x line.

Let f have a domain A range B. If f is one-to-one, then the inverse function of f is called f^-1 where the domain of f^-1 is B, and the range is A.

Question 18

find inverse

Answer 18

Write y=f(x)
Manipulate y=f(x) until you get f^-1(y)=x
Write y=f^-1(x)

If og function is not one-to-one (eg. parabola) have to restrict domain , making it one-to-one → x>=0 OR x<=0

Question 19

logex

Answer 19

ln(x)

Question 20

logb (xy)

Answer 20

= logbx + logby

Question 21

logb(x/y)

Answer 21

= logbx - logby

Question 22

logb(x^r)

Answer 22

= rlogb(x)

Question 23

ln(e^x))

Answer 23

=x

Question 24

e^ln(x)

Answer 24

=x

Question 25

ln(x)

Answer 25

= loge(x)

Question 26

e^rln(x)

Answer 26

= e^ln(x^r) = x^r

Question 27

lnx=y

Answer 27

e^y = x

Question 28

ln(e)

Answer 28

1

Question 29

ln(e^2)

Answer 29

2

Question 30

ln(e^3)

Answer 30

= 3

Question 31

ln(xy)

Answer 31

lnx+ lny

Question 32

ln(x/y)

Answer 32

ln x - lny

Question 33

ln x^y

Answer 33

y ln x

Question 34

ln(1/x)

Answer 34

- ln(x)

Question 35

ln as log

Answer 35

log base e = ln