Inverse Function Properties
Explore the properties of inverse functions, logarithms, and integrals, along with examples and formulas. Learn about the natural logarithm, derivatives, and integrals of trigonometric functions like tan, cot, sec, and csc.
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Second semester Inverse function Suppose that is a one to one function on a domain D with rang R. The inverse function ? 1 is Defined by ? 1? = ? ?? ? ? = ? The domain of ? 1is R and the range of ? 1 is D. Note: The symbol ? 1 for the inverse of ? is read ? inverse. The -1 in ? 1 is not an exponent; ? 1(x) dose not mean 1 ?(?) . Notice that the domain and range of ? and ? 1 are interchanged. ? ???? ? 1(?) Example: Let ? ? = ? 2 ? ? = ? 2 ?? 2? = ? 2? ? ? 1 = 2? ? = ? 1 2? ? 1 Replace x by y to obtain ? = 2? is ? 1(?) ? = ? 1
Example: Find the inverse of the function ? = ?2 , ? 0 expressed as a function of x ? = ?2 ? = ? ??????? ? = ? To calculate the derivatives of ? = ?2 ? ?? ? ? = 2? ? ??? 1= 1 1 , ? = ?(?) 2 ?= 2 ? 1(?) 1 ? (?) ? 1 ? = Or ?? 1 ?? 1 = ?? ???=? 1(?) ?=?
??1 ?? ?? ? = 6 = ? 2 ??? ??? ??????? Example: Let ? ? = ?3 2 .???? ? ? ????? ?? a formula for ? 1? . ?? 1 ?? 1 1 1 = = 3(2)2= ?? ???=2 12 ?=?(2) Natural Logarithms The natural logarithms is the function given by ? 1 ln(?) = 1 ??? , ? > 0 ?1 ln? = ??? 1 ?1 ? ??ln? = ? ?? ??? =1 ? 1
Definition: The number e is that number in the domain of the natural logarithm satisfying ln ? = 1 ,? = 2.71828 So the number e lies within the interval (2, 3) and satisfies ?1 ??? = 1 ? ??(ln?) =1 ?? ?? ? > 0 ? 1 Properties of logarithms ln?? = ln? + ln? ln? ?= ln? ln? ln1 ?= ln? ln??= ??? ? ? ??? The integral If u is differential function that is never zero 1 ??? = ?? ? + ?
Example: Evaluate 2 2? ?2 5?? 0 ? = ?2 5 1?? ?= ?? = 2??? 1 ln? = ln 1 ln 5 5 5 = ln1 ln5 = ln5 Example: Find ?2???3? cos?41 + ? ?? ?? ?? ? = ln ? = ln ?2???3? ln(cos?41 + ?) = ???2+ ln???3? lncos?4 1 2ln(1 + ?) ?? ??=2 =2 3 cos?4( sin?4) 4?3 1 1 1 ?+ ?+ 3cot? + 4?3tan?4 sin?cos? 2 1 + ? 1 2 1 + ?
Example: Find ?? ?? ??? ? = ?sin ? ln? = sin?ln? ?? ??= sin?.1 1 ?sin? + ln? cos? = ?sin ?1 ?sin? + ln? cos? 1 ? ?+ ln? cos? ?? ??= ? cos(ln ?) ? ?? Example: Evaluate ?? =1 ? = ln? ??? cos? ?? = sin? + ? = sin(ln?) + ?
The integrals of tan x, cot x, sec x, and csc x tan? ?? = sin? cos??? = ?? , ? = cos? ?? = sin? ?? ? = ?? ? + ? = ln cos? + ? 1 cos?+ ? = ln sec? + ? = ln cot? ?? = cos? sin??? = ?? ?= ln ? + ? = ln ???? + ? , = ln ? = sin? ?? = cos? ?? 1 csc?+ ? = ln csc? + ? 1 1 + ? = ln sin? sec? ?? = sec?sec? + tan? sec? + ?????? = ???2? + sec?tan? sec? + tan? ?? ,? = sec? + tan? ?? = (sec? tan? + ???2?)?? = ?? ?= ln ? + ? = ln sec? + tan? + ?
csc? ?? = csc???? ? + cot? csc? + cot??? = ???2? + csc? ???? csc? + ???? ?? ,? = csc? + ???? ?? = ( csc? ???? + ???2?)?? = ?? = ln ? + ? = ln csc? + cot? + ? ? Exponential functions , ??? ? = ?? The function ln x, being an increasing function of x with domain (0, ) and range ( , ) , has an inverse ?? 1? with domain ( , ) and range (0, ). The function ?? 1? is usually denoted as ??? ? = ??. ?? 1? = ?? ?ln ?= ? ??? ??? ? > 0 ln ??= ? ??? ??? ?
Example: Solve the equation ?2?6= 4 ??? ? ln?2? 6= ln4 2? 6 = ln4 2? = ln4 + 6 ? =ln4 + 3 = ln41/2+ 3 2 = ln2 + 3 ? ????= ??.?? ???? = ??+ ? ?? , Example: ? ?? ?sin ?= ?sin ?.cos? ln 2?3??? Example: Evaluate 0 ? = 3? ?? = 3?? ??8 ln 8 =1 1 3?? =1 ? = 3ln2 ? = ln23= ln8 ???? = ?ln 8 ?0 3 3 0 0 =1 38 1 =7 3
? 2?sin ?.cos? ?? Example: Evaluate 0 ? 2???? = ,? = sin? ?? = ???? ?? 0 1= ?1 ?0= ? 1 = ?? 0 ???? ?? ????????? 1 1 - ??1.??2= ??1+?2 2. ? ?= ?? 3. ??1 ??2= ??1 ?2 ?????????? 1. ??? ??? ?????? ? > 0 ??? ? ,? ? ??????????? ???????? ??? ???? ? ?? ??= ????? 2.??? ??? ? > 0 ??? ??? ??? ???? ?????? ? 4. ??1 ?= ???1 ??=?? ln ? The derivative of ?? To find this derivative, we start the defining equation ??= ???? ? ? ?????? ?= ???? ?.? ?? ? ?? ??= ??? ? = ???? ?.ln? = ??.ln?
