math 143 exam 2

difference quotient

f(x+h)-f(x) / h when h cannot equal 0
definition of derivative

f(x) = ⌊x⌋

round down to nearest integer
(ex. 4.3 = 4)

f(x) = ⌈x⌉

round up to nearest integer
(ex. 4.3 = 5)

(f+g)(x)

f(x) + g(x)

(f-g)(x)

f(x) - g(x)

(fg)(x)

f(x)g(x)

(f/g)(x)

(f)(x)/g(x)
g cannot equal 0

(fog)(x)

f(g(x))

domain of (fog)(x)

has to be in the inner function AND (fog)(x)

inverses

function has to be one-to-one (pass horizontal line test)

how to find inverse

- swap x and y
- solve for y
f-1(y) = x
f(x) = y

inverse domain and range

f(x): D: A, R: B
f-1(x): D: B, R:A

when are functions inverses?

f(g(x)) = g(f(x)) = x

standard form

ax^2+bx+c
c is y int
vertex: (-b/2a, plug (-b/2a) back into function)

vertex form

a(x-h)^2+k
vertex: (h,k)

x-intercept form

a(x-e)(x-d)
(e, 0) (d,0) are x-intercepts

end behavior (odds)

+x^0
arrow down x arrow up

-x^0
arrow up x arrow down

end behaviors (even)

+x^0
both arrows up
-x^0
both arrows down

multiplicity of zeros

even: bounce
odd: through

ration functions: domain

values of x that make denominator 0

ration functions: vertical asymptote

in reduced function, values of x that make denominator 0

rational functions: x-intercept

values of x that make numerator 0 (plug in 0 for y and solve)

rational functions: y-intercept

plug in 0 for x and solve

rational functions: holes

in the thing that cancels, x value that makes the canceled item 0

horizontal asymptote rules

ax^n / bx^m
nn=m | ha: y=a/b
n>m | ha: none

directly proportional

y=kx

inversely proportional

y = k/x

a^x-a^y

a^x+y

a^0

1

(a^x)^y

a^xy

a^-x

1/a^x

f(x) = ab^x

b >1 = growth
0 < b

(ab)^x

(a^x)(b^x)

a^m / a^n

a^m-n

a^m / b^m

(a/b)^m

f(x) = 1^x

D: all reals
R: y>0
HA: y=0

loga(a)

1

loga(a)^x

x

log(a)1

0

a^loga^x

x

logaMN

logaM + logaN

logaM/N

logaM - logaN

logaM^r

rlogaM

logaM

logb^M/logb^a

f(x) = log x

D: x>0
R: all reals
VA: x>0