MA108 CALCULUS WITH ALGEBRA I
Tuesday, 10/2/12
Today:
HW #2 is posted on our website
Exponential growth and decay
Inverse functions
Reading:
Exercises (not to hand in):
1.5, 1.6
1.5: 29
1.6: 1-7 odd, 19, 21, 35, 37
Tuesday, 10/2/12, Slide #1
Example: Exponential Growth & Decay
Example: Suppose we start with a culture
of 100 bacteria, and every hour the number
of bacteria doubles. Write a formula for the
number of bacteria after t hours have
passed.
Exponential Growth/Decay: A population
whose growth/decay rate is proportional to
the population. If r > 0 is the growth rate:
P(t) = P(0) rt
Growth if r > 1, decay if 0 < r < 1.
Tuesday, 10/2/12, Slide #2
New Functions from Old:
Inverse Functions
Inverse functions: If a function can be
reversed (i.e., the output becomes the input
and the input becomes the output), the
reverse function is called the inverse of the
original.
Example: The inverse of f(x) = x3 is the
cube root function: g ( x ) = 3 x
Example: What about f(x) = x2 ?
Tuesday, 10/2/12, Slide #3
Inverse Functions: Definitions and
Notation
Two functions f(x) and g(x) are inverses of each
other if the following identity holds:
f(a) = b ¤ g(b) = a,
for any a, b for which f(a) and g(b) are defined
To have an inverse, a function must be one-toone, meaning no two different inputs have the
same output: If f(v) = f(w), then v equals w.
This means f passes the horizontal line test
Notation: If g(x) is the inverse function of f(x),
then we write g(x) = f-1(x).
This does not mean 1/f(x)!!
If f(x) = x3, then: f −1 ( x) = 3
x
Tuesday, 10/2/12, Slide #4
Finding formulas for inverses
If y = f(x) is one-to-one, how do we find
its inverse? Use the definition:
y = f(x) ¤ x = f-1(y)
Step 1: Write y = f(x).
Step 2: Solve this equation for x.
Step 3: If there is a unique solution, then that
solution is f-1(y).
Step 4: We usually rewrite the inverse using
the letter ‘x’.
Example: Find the inverse function for:
f(x) = 7 x5 – 12.
Tuesday, 10/2/12, Slide #5
Exercise
Find
a formula (as a function of x )
for the inverse of f(x) = (2x +1)3.
A. y = (2x +1)1/3
B. y = (2x +1) -3
C. y = (x 1/3 – 2) + 1
D. y = (x 1/3 – 1) / 2
E. y = 2x 1/3 - 1
Tuesday, 10/2/12, Slide #6
Logarithmic Functions
Log functions are the inverses of
exponential functions:
If f(x) = bx, then f-1(x) = logb(x)
y = bx ñ x = logb(y)
In other words:
Evaluate:
logb(x) is the exponent on b that gives x.
log5(125) (Ask 5 to what exponent equals 125?)
log8(2)
É log3(◊3 )
log 4 (4 76 )
É
9 log9 ( 231)
Tuesday, 10/2/12, Slide #7