Fall 2004, Triginometry 1450-02, Week 4-5

Chapter P8. P9Composition and Inverse of Functions / DAY 9
pp.94-109

1. General Picture

A function: or .

One element from the domain goes to only one element from the range.

A domain is the set of all possible values (inputs) of for which the function is defined.

A range is the set of all possible values (outputs) of .

Composition ,
(function of functions) / /
Identity function /
Inverse function /
/

EXAMPLE 6.p95Compositions of Functions

and . Find the values.

Solution:

/ 1. Always write this notation
/ 2. Then substitute
/ 3. Then put
/ 4. Put in the new function
Solution is: 2 / Write the last answer.
REMEMBER:
  1. Remember:
  2. Then substitute .
  3. Then put

EXAMPLE 8. p109Inverse of a Function

Find the inverse of .

Solution:

/ 1. Write the function. Find from this function
/ 2. Interchange and (after finding)
/ 3. Write
Try to draw the graph of 2 functions
Solution is: / Write the last answer.
REMEMBER:
  1. Find from the original function
  2. Interchange and
  3. Write

  • Finding the inverse means to find from the original function.
example / ELEMENT / Identity / INVERSE / example
/ / 1 / /
/ Square / 0, 1 / Square root /
/ Exponent / / Logarithm /
axis / 0 origin / axis
Domain / Range

2. Existence of the Inverse Function

DEFINITION:
A function: , ,
Then is the inverse function of if
  1. for every in the domain of and
  2. for every in the domain of .
A functionis one-to-one if in its domain,
  • implies that
A functionhas an inverse functionif and only if is one-to-one.
  • To check one-to-one condition needs to check
  • See the example 6 pp107

EXAMPLE 3. p105Verifying Inverse Functions

Show that they are inverse functions. and and .

Solution:

/ Always write this form.
since / Substitute and check the condition 1.
/ Substituteand check the condition 2.
Solution is: there are inverse functions / Write the last answer.
REMEMBER:
  1. Use
  2. Check 2 conditions substituting two functions.

Chapter P7. P8 Exponential and Logarithmic Function / DAY 10
pp.445-464

1. General Picture

DEFINITION:
is called the exponential function with base.
  • where and . Domain: Range:
is called the natural exponential function with natural base.
  • The number is named after the Swiss mathematician Euler (1707-1783). . This number is called the natural base.

DEFINITION:

is called the logarithmic function with base. (Read: log base of )
  • where and and . Domain: Range:
is called the natural logarithmic function with natural base.
  • The number is called Euler number.
is called the common logarithmic function with base10.
  • Four Arithmetic operations and Exponentation (repeated multiplication)
Operation / Addition / Subtraction / Multiplication / Division / Exponentiation
Algebraic expressions / / / / , where /
sum / difference / Product / Quotient / Exponent
An exponent is thepowerin an expression of the form. The process of performing the operation of raising the base to the poweris known asexponentiation.
  • Real numberscontain natural, integer, rational, and irrationalnumbers.
Exponent / Zero exponent / Natural exponent / Negative exponent / Rational (radical) exponent / Irrational exponent
Number / / / / /
Value / / (factor times) / / /
SPECIAL PROPERTIES OF EXPONENTS
1.
zero exponent / 2.
negative exponent / 3.

Radical exponent / 4.
Radical exponent
5. Product rule / 6.
Power of power rule / 7. Power rule / 8. quotient rule

Exponential and Logarithmic Function

Exponential function / Inverse function / Logarithmic function

, , / /
, and
Domain:
Range: / Domain, range switched / Domain:
Range:
1 / / Zero exponent /
2 / / Unit exponent /
3 / / Identity, inverse property / =
4 / / One-to-one property /
5. Product
rule / / Product of 2
Sum of exponents /
6. Quotient
rule / / Quotient of
Difference of exponents /
7. Power
rule /
power of power / Power of power product /
Power falls down
8. Change of Base rule / / /

Proof of 3: Let .. After taking the logarithm from with base ,

we get by the definition. Also, from the definition we get .

Proof of Product: Let and. Then and.

So . After taking the logarithm with base ,.

Hence, .

Proof of Power Rule: Let . Then . So . After taking the logarithm

with base ,using 3. Hence, .

Proof of Change of Base Rule: Let . After taking the logarithm with base ,

. Hence, .

2. Exponential function

EXAMPLE 9. p453Compound interest

Find the balance after 4 years if the interest is compounded (a) quarterly and (b) continuously. A total of 12000$ is invested at an annual interest rate of 3%.

Solution:

. For compoundings per year: . For continuous compounding per year: . Substitute . Then .

a) for quarterly compoundings, . So after 4 years at 3%, the balance is =

(b) For continuous compounding, the balance is

3. Logarithmic function

is called the logarithmic function with base. (Read: log base of )

  • where and and . Domain: Range:

EXAMPLE2, 3. p459Logarithmic Properties

See the examples. Calculate using LOG function and try to proof general properties.

Draw the graphs.

EXAMPLE 9. p463Domain of Logarithmic Functions

Find the Domain.

Solution:By the definition of logarithm, and . So domain is:

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Batmunkh.Ts Math Graduate Student 09.19.2004