Transcript 4.1 Composite and inverse functions

4.1 Composite and inverse functions
composite
f(x) = 2x – 5
g(x) = x2 – 3x + 8
(f◦g)(7) =
decomposing
h(x) = (2x -3)5
(f◦g)(7) = f(g(7))
= f(72 – 3∙7 + 8)
= f(36)
= 2∙36 – 5
= 67
f(x) = x5
g(x) = 2x – 3
(f◦g)(x) = (2x – 3)5 = h(x)
f(x) = ?
g(x) = ?
inverse relation
f(x) = x2 – 6x
f-1(x) = ?
y = x2 – 6x (switch x and y)
x = y2 – 6y
x + 9= y2 – 6y + 9
(x + 9) = (y - 3)2
(x + 9)½ = y – 3
(x + 9)½ + 3= y = f-1(x)
f(x) = 5x + 8
f-1(x) = (x-8)
5
(f-1◦f)(x) = f-1(f(x)) =
= f-1 (5x + 8) =
= (5x + 8) – 8
5
= 5x = x
5
(f-1◦f)(x) = ?
4.2 Exponential Functions and Graphs
Compound Interest
A is the amount of money
that a principal P will be
worth after t years at an
interest rate of i,
compounded n times a year.
A = P(1 + i/n)nt
y = 2x
y=x
x = 2y
$100,000 is invested for t
years at 8% interest
compounded semiannually.
e
A = $100,000(1 + .08/2)2t
A = $100,000(1.04) )2t
t= 0 A = $100,000
t= 4 A  $136,856.91
t= 8 A  $187,298.12
t=10 A $219,112.31
2.7182818284...
4.3 Logarithmic Functions and Graphs
For any exponential function
f(x) =ax, it inverse is called a
logarithmic function, base a
f(x)=2x
f-1(x) = log2x
f(x)=2x
f(x) = x
f-1(x) = log2x
Write x = ay as a
logarithmic function
loga1 = 0
and
logaa = 1
for any logarithmic base a
logax = y
if the base is 10
then it is called
a common log
4.3 Logarithmic Functions and Graphs (cont)
f(x)=ex
For any exponential function
f(x) =ex, it inverse is called a
natural logarithmic function
f-1(x) = ln x
logbM = log M
The Change of base
formula
a
logab
Write x = ey as a
logarithmic function
loge1 = ln 1 = 0
and
logee = ln e = 1
for any logarithmic base e
ln x = y
if the base is e
then it is called
a natural log
4.4 Properties of Logarithmic Functions
Product Rule
logaMN = logaM + logaN
Power Rule
logaMp = plogaM
The Quotient Rule
Logarithm of
a Base to a Power
logaM/N = logaM - logaN
loga ax = x
4.4 Properties of Logarithmic Functions cont.
A Base to a
Logarithmic Power
alogax = x
loga75 + loga2
loga150
ln 54 – ln 6
ln 9
5 log5(4x-3)
4x - 3
4.5 Solving Exponetial and Logarithmic Equations
For any a>0, a1
Base – Exponent
Property
ax = ay
x=y
23x-7 = 25
3x-7 = 5
3x = 12
x=4
3x = 20
log 3x = log 20
x log 3 = log 20
x = log 3 / log 20
x  2.7268
ex – e-x – 6 = 0
ex + 1/ex – 6 = 0
e2x + 1 – 6ex = 0
e2x – 6ex + 1 = 0
ex = 3 8
ln ex = ln (3 8)
x = ln (3 8)  1.76
4.6 Applications and Models: Growth and Decay
Exponential growth
Population
P(t) = P0ekt
where k>0
In 1998, the population of
India was about 984 million
and the exponential rate of
growth was 1.8% per year.
What will the population be
in 2005?
P(7) = 984e0.018(7)
P(7)  1116 million
Interest Compounded
Continuously
P(t) = P0ekt
$2000 is invested at an
interest rate k, compounded
continuously, and grows to
$2983.65 in 5 years. What is
the interest rate?
P(5) = 2000e5k
$2983.65 = $2000e5k
1.491825 = e5k
ln 1.491825 = 5k
k  0.08 or 8%
4.6 Applications and Models: Growth and Decay cont.
kT = ln2
Growth Rate and
Doubling Time
k = ln2/T
T = ln2/k
Logistic Function
Models of Limited Growth
P(t) =
a .
1 + be-kt
Exponential Decay
P(t) = P0e-kt
where k>0
Converting from
Base b to Base e
bx = e x(lnb)