Two really important ideas
Function Inverse
&
Exponential Function
Function Inverse
Going Driving
I start 10 miles away from my house and drive
away from my house at 30 mph. If I know how
long I’ve been driving, how far am I from my
house?
Going Driving
I start 10 miles away from my house and drive
away from my house at 30 mph. If I know how
long I’ve been driving, how far am I from my
house?
d=number of miles away from my house
t=number of hours I’ve been driving
d=30t+10
Going Driving
I start 10 miles away from my house and drive
away from my house at 30 mph. If I know how
far I am from my house, how long have I been
driving?
d=number of miles away from my house
t=number of hours I’ve been driving
d=30t+10
Going Driving
I start 10 miles away from my house and drive
away from my house at 30 mph. If I know how
far I am from my house, how long have I been
driving?
d=number of miles away from my house
t=number of hours I’ve been driving
d=30t+10
(d-10)/30=t
Going driving
I haven’t changed anything
(except my point of view)
d=30t+10
(d-10)/30=t
I haven’t changed anything
(except my point of view)
40 miles
1 hours
40 miles
1 hours
10 miles
0 hours
0 hours
d=30t+10
10 miles
(d-10)/30=t
Cubing
I have an equation y=x3. I know x=2 and I want
to figure out y.
y=(2)3
y=8
I have an equation y=x3. I know y=8 and I want
to figure out x.
8=(x)3
y=∛8=2
Cubing to cube root
Cubing to cube root
y=x3
x=∛y
Cubing to cube root
The relationship between x and y stays the same
Only my point of view changes
y=x3
x=∛y
Notation
From x  y
y=x3
ƒ(x)=x3
Notation
From x  y
y=x3
ƒ(x)=x3
From y  x
y=x3
∛y=x
∛y=ƒ-1(y)
ƒ-1(y)=∛y
Notation
From x  y
y=x3
ƒ(x)=x3
f
x® y
f
-1
x¬ y
From y  x
y=x3
∛y=x
∛y=ƒ-1(y)
ƒ-1(y)=∛y
Notation
From x  y
y=x3
ƒ(x)=x3
Because x and y
don’t actually
mean anything, I
can change their
names if I want.
From y  x
y=x3
∛y=x
∛y=ƒ-1(y)
ƒ-1(y)=∛y
ƒ-1(x)=∛x
Notation
From x  y
y=x3
ƒ(x)=x3
Because x and y
don’t actually
mean anything, I
can change their
names if I want.
From y  x
y=x3
∛y=x
∛y=ƒ-1(y)
ƒ-1(y)=∛y
ƒ-1(x)=∛x
This is not actually a
good idea, but it’s
popular in many
math books
How to find a function inverse
•
•
•
•
•
ƒ(x)=………….x………….
Rewrite as y=……………x…………
Solve for y. x=~~~~y~~~~~~
Rewrite as an inverse ƒ-1(y)=~~~~y~~~~~~
OPTIONAL: change ys to xs.
• ƒ-1(x)=~~~~x~~~~~~
• WARNING: Always check that your inverse is
actually a function.
Round trip
I drive away from home for 1.25 hours at 30
miles per hour, then I turn around and drive
back home at 30 miles per hour.
y=number of miles I am from home
x=number of hours since I started driving
Round Trip
Round Trip
If I know x (time), I can
figure out y (distance).
y is a function of x.
If I know y (distance),
I can’t figure out y (time).
x is NOT a function of y.
Testing if the inverse is a function
A shoe size that is size ‘x’ in the United States is size
t(x) in Continental size, where t(x)=x+34.5 Find a
function that will convert Continental shoe size to a US
shoe size.
A)
B)
C)
D)
E)
t-1(x) = 1/(x+34.5)
t-1(x) = 1/x + 34.5
t-1(x) = 34.5 + x
t-1(x) = x – 1/34.5
None of the above.
A shoe size that is size ‘x’ in the United States is size
t(x) in Continental size, where t(x)=x+34.5 Find a
function that will convert Continental shoe size to a US
shoe size.
t(x)=x+34.5
y=x+34.5
y-34.5=x
y-34.5=t-1(y)
t-1(y)=y-34.5
t-1(x)=x-34.5
E
Exponential Functions
The “I’m going to lie to you a lot” version
Exponential functions measure
steady growth
• If you really want to know what that means
exactly, take differential equations (after
Calculus)
• Here’s the basic (lying) version
• An exponential growth happens when
something is making more of itself (in a
“steady” way)
• People, money, bacteria, etc…
Example
• One dollar makes one dollar every year.
$1
Year 0
$1
Year 1
Example
• One dollar makes one dollar every year.
$1
$1
Year 0
$1
Year 1
Example
• One dollar makes one dollar every year.
$1
Year 0
$1
$1
$1
$1
Year 1
Year 2
Example
• One dollar makes one dollar every year.
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
Year 2
Example
• One dollar makes one dollar every year.
$1
$1
Year 0
$1
Year 1
$1
$1
$1
$1
$1
$1
$1
$1
Year 2
Year 3
Example
• One dollar makes one dollar every year.
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Example
• Every year I keep what I have and add what I have.
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Example
• Every year I double my money
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Example
• Every year I double my money
$1
x
y=1(2 )
y=# of $
x=# of yrs
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
$1
Year 0
$1
Year 1
$1
$1
Year 2
Year 3
Exponential Growth
Exponential Decay
Which of the following functions represent that of
exponential decay?
A)
B)
C)
D)
E)
x
f(x)=(1/2)
-x
f(x)=(1/2)
-x
f(x)=(1/3)
(b) and (c)
None of the above
Which of the following functions represent that of
exponential decay?
A)
B)
C)
D)
E)
x
f(x)=(1/2)
-x
f(x)=(1/2)
-x
f(x)=(1/3)
(b) and (c)
None of the above
Anatomy
•
•
•
•
•
•
•
•
The standard form of the exponential is y=abx
a is called the initial value (y-intercept)
b is called the growth factor.
When 0<b<1, you have exponential decay
A non-standard form is y=ac-x
a is the initial value (y-intercept)
c-1=1/c is the growth factor.
When 0<1/c<1, you have exponential decay
The compound interest formula
• P dollars are invested at r% per year
compounded n times per year. After t years, I
have A dollars.
nt
æ
rö
A = P ç1+ ÷
è
nø
The compound interest example
• 100 dollars are invested at 7% per year
compounded monthly. How many dollars do I
have after 5 years?
• P=100, r=7/100, n=12 (12 months a year), t=5
Find A.
nt
æ
æ .07 ö
rö
A = P ç1+ ÷ = 100 ç1+
÷
è
è
nø
12 ø
12(5)
The compound interest example
• I invested some money at 3% per year
compounded quarterly. After 9 years I had
$1000. How much did I start with?
• R=3/100, n=4, t=9, A=1000, find P.
nt
æ
rö
A = P ç1+ ÷
è
nø
æ .03 ö
1000 = P ç1+
÷
è
4 ø
4(9 )
Asymptotes
• All exponentials y=abx have asymptote y=0
Asymptotes
• Adding c moves a graph up by c.
Asymptotes
• Any function y=abx+c has asymptote y=c
Consider the function below:
Which of the following statements matches with this
function?
a)
b)
c)
d)
e)
As x approaches infinity, f(x) approaches 0.
As x approaches negative infinity, f(x)
approaches 0.
As x approaches infinity, f(x) approaches -4.
As x approaches negative infinity, f(x)
approaches -4.
None of the above
D) As x approaches negative
infinity, f(x) approaches -4.