1.6 Inverse Functjons and Logarithms
One-to*oile Functions: eachy-value of a function
corresponds to exactly one x-value
Pass
Concepts to know #
2
MATH 131
Over Ch 1.5,1.6 ,2"\
1.5 Exponent!4 Function-s:
t
il rost
/{t}= fr',f,:} 0, a
*1
Inverse Functions: J:-l
Be able
to find the inverse of a 1-1. function.
Know that the domain and range
range and domain of
Exponential growth when a > I
Exponential decay when 0 < a < I
Domain: (-oo, oo), Range: (0,
both vertical and horizontal line tests
Have inverses
Find the inverse of
*)
.f
of f-' are the
,
jT 1-l:
/(x)' : Zx+l
Horizontal Asymptote; .P = 0 in one direction
Know the shape of the basic exponential function
fcno* exponentiat properties in order to simpliry
an expression, or to solve an equation'
q**v
=e, .a.v,a,*! =$,{o')t
Logarithmic Functions
= au,(ob)* = frrb'
Basic tunctio n:
f (t)
= loga x ,b
>' 0o b
*
1
Inverses of exponential functions
Domain: (0,
*) , Range: (**, *)
Base e is natural
log: ln x
Base 1.0 is cornmon
log: log x
Know basic shaPe of the graph
Knowpropertiesoflogarithmsinorderto
simpliff expressions and solve equations
Properties
le to
e
wo
transformations' and
into an exponential
s, use
function.
b'*
Laws of Logarithms
Known for th-eirquick reproducdve cycle, female
and
rabbits, called does, are capable sf producing
the
If
year.
a
in
individuals
weaning up to 60
every
on the isla
poputati
Y there we
4 month
fter Z
a. What is the size of th
years?
b. wrat is the size of the population
after
months?
c.Whatisthesizeofthepopulationafter
17 months?
d.
loga
lWol = 0
logrb=L
After how many months does the
rabbits?
PoPulation reach 1,000
t
logAB = lo# +
.A
lsgB log; :
ogA
logAr = PlogA
to expand using the laws
Be able
to combine several log
expresslons into a single logarithm'
Be able
* logB
=x
Change of base fornrula
:I'-Sd
It.rg,, tt
log,,
a. Simpliff' logt 8"
b. Simpliff:
logo
c* Simpliff:
logffi
e.
3
simpify
Be able to
d.
192- logu
logr 81
"*s
f hrlJ
\t)
to solve;
Be able
g. log,g=1
2
h, lnx*-3
i. log* Ji = x
Be able to
find the domain:
i. h(x) = logs (8 *
h s@)=
r.
if(*)
=
ln(x
2*)
- x')
Jm-
lo*s (10-
x)
Write as a single logarithm:
rn. log rc' -log 3y + log Zzj
,
Z,L Tangent and Velocity Problems
Know how to find the average rate of change
between
two points;
Av
C
,
slope of a secant
line
Average velocity: change in position per change in
Ld
time #
At
Be able to estimate the instantaneous velocity
or slope of the tangent line using numerical
approximations {choosing values closer and
clcser to a from the right and leftJ-
The point P (1,4)ties on the graPh of
y -ffi
+2.
Estimate the slope of the
tangent to the function at x = 1 using numerical
approximations, Show Your work.