2. Any function where a > 0,
b > 0, and b ≠ 1 is called an
exponential function with base b.
Domain: All Real Numbers
Range: f(x) > 0 (positive real #s)
3. If f is an exponential function,
f(0) = 3, and f(2) = 12, find f(-2).
Solution:
Substitute given values for x and f(x)
into :
f(0) = 3 3 = ab0
b0 = 1, so a = 3
f(2) = 12 12 = 3b2
So b = 2
Therefore, f(x) = 3 ∙ 2x and f(-2) = ¾
4. If h is an exponential function such
that h(0) = 5 and h(1) = 15, find h(x).
Answer:
5. Remember, exp. growth and
decay functions can be written
A(t) = A0 (1 + r)t (Form 1)
Can also be written A(t) = A0bt/k
(Form 2)
where
k = time needed to multiply A0 by b
6. A bank states you can double
your money in a savings account
in 12 years. Express A(t) using
both forms. What is the interest
rate?
Solution:
(Form 2) A(t) = A0bt/k k = 12, b = 2
A(t) = A02t/12 Notice: A(12) = A0212/12 = 2A0
(Form 1) A(t) = A0(21/12)t = A0(1.059)t
A(t) = A0(1 + 0.059)t
r = 0.059 Interest rate = 5.9%
7. You invest $1000 in an
account and your money triples
in 15 years. What is the
interest rate to the nearest
tenth of a percent?
Answer: 7.6%
8. A radioactive isotope has a half-life
of 5 days. At what rate does
the substance decay each day?
Solution:
A(t) = A0bt/k k = 5, b = 1/2
r = 0.13 Rate = 13%
9. If a quantity is growing at rate r%
per year (or month, etc.) then the
doubling time is approximately
(72 ÷ r) years (or months, etc.)
For example, if a quantity grows
at 8% per month, its doubling time
will be about 72 ÷ 8 = 9 months.
10. An investment grows at a rate
of 10% per year. About how
long will it take to double your
money?
Solution:
Use “Rule of 72”: (72 ÷ 10) = 7.2
So, about 7 years.
