PreCalc 5.1-5.6

Test for precalc!!!!

A _______ is a relation if for every x there is exactly one y

function. (can be determined by vertical line test

Interchanging the x and y coordinates of each order pair in a relation produces the _______

inverse relation

If a relation is defined by and equation ________ produces an equation of the inverse relation

switching the variables

A function f is __________ (or injective) if f(a)= f(b), then ___________

one-to-one a=b

A function is one-to-one if ____________

for every output there is exactly one input

The _______ of a function f(x) is denoted _______

inverse f^-1(x)

How to find an inverse?

1. Call f(x)=y

2. Switch x and y

3. Solve for y and call it f^-1(x)

Verifying f^-1(x)

1. Check compositions: f(f^-1(x))=x

2. Graphically: The graphs of a relation and its inverse are always reflections of each other across the line y=x

If f(x) is not one-to-one it ________ still have an inverse….but wont be a function unless we ________ the domain of f

can restrict

The function _________, where x is a real number, a>0 and a≠1, is called the ________, base a.

f(x)=a^x exponential functions

Common characteristics of f(x) =a^x

1. continuous

2. one-to-one

0<a<1

decreasing

a>1

increasing

Domain

Range

(-∞, ∞)

(0, ∞)

Horizontal Asymptote

y=0

y-intercept _____. Since _______, regardless of a

(0,1) a^0=1

Translations are ________ as other functions

the same

Compound Interest Formula

A(t) = P( 1 + r/n) ^nt

A(t)

Amount after t years

P

principal (starting amount)

r:

Interest rate (decimal)

n

Number of times per year that interest is compounded

Annually_____, Quarterly_______,Semiannually_______, Daily________

1,4,2,365

In f(x)= e^x, e=

2.72

f(x) = e^x is _______ and _______

continuous and one-to-one

f(x) = e^x

Domain

Range

(-∞,∞)

(0, ∞)

f(x) = e^x

Horizontal Asymptote

y-intercept

y=0

(0,1)

The function _______ where x is a real number, a>0 and a≠1, is an exponential function.

f(x)^a

f(x)^a- This function is one-to-one……so, it has an __________.

inverse function

Natural Logarithm

f(x)= lnx

x^a x x^b

= x^a+b

x^a/x^b =

x^a-b

(x^a)^b =

x^ab

x^-p

1/x^p

n√x^m =

x^m/n

The product rule

loga(MN)= logaM+logaN

the quotient rule

logaM/N=logaM-logaN

The power rule

logaX^p=ploga^x

logaMN ≠

(logaM)(logaN)

logaM/N ≠

logaM/logaN

(logaX)^p≠

logaX^p

logaA^x =

x

ln(e^x)=

x

aloga^x=

x

e^lnx=

x

y= logaX ⟺

a^y = x

y = lnx ⟺

e^y =x

logaA^x = ____ and a^logaX = ________

x x

ln(e^x)=________and e^lnx=_______

x x

Base exponent property

for any a>0, a≠1

property of logarithmic equality

for any M, N >0, a>0, a≠1

General form for exponential growth and decay is

P(t)= P0e ^kt, k>0

P(t)=

the amount after time t

P0

is the initial amount

exponential ______, and k is called the growth rate

growth

exponential ________, and k is called the decay rate

decay