Inverse Functions
Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) 1 E: f 5 and f 5 are inverses because their input and output are switched. For instance: f 4 f 1 4 4 5 f f ' 5 4
Tables and Graphs of Inverses Orginal Switch and y Inverse (0,5) (,16) (6,4) (10,0) (0,5) (18,16) (14,4) X Y 0 5 16 6 4 10 0 14 4 18 16 0 5 X Y 5 0 16 4 6 0 10 4 14 16 18 5 0 (4,14) (0,10) (4,6) Line of Symmetry: y = Switch and y (16,18) (16,) Although transformed, the graphs are identical
Inverse and Compositions In order for two functions to be inverses: f g AND g f
One-to-One Functions A function f() is one-to-one on a domain D if, for every value c, the equation f() = c has at most one solution for every in D. Or, for every a and b in D: f a f b a b unless Theorems: 1. A function has an inverse function if and only if it is one-to-one.. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.
The Horizontal Line Test If a horizontal line intersects a curve more than once, it s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
The Horizontal Line Test If a horizontal line intersects a curve more than once, it s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
Eample Without graphing, decide if the function below has an inverse function. f 6 3 If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. See if the derivative is always one sign: f ' 6 Since the derivative is always negative, the inverse of f is a function.
Find the Inverse of a Function 1. Switch the and y of the function whose inverse you desire.. Solve for y to get the Inverse function 3. Make sure that the domains and ranges of your inverse and original function match up.
Eample Find the inverse of the following: d 4 3 Switch and y Only Half Parabola Solve for y 34 4 y 3 y Really y = Restrict the Domain! 3 Full Parabola y (too much) 4 d 3 y 4 1 3 4 when 3 =3 Make sure to check with a table and graph on the calculator.
Logarithms v Eponentials
Definition of Logarithm The logarithm base a of b is the eponent you put on a to get b: lo g a b a > 0 if and only a i.e. Logs give you eponents! b if and b > 0 The logarithm to the base e, denoted ln, is called the natural logarithm.
Logarithm and Eponential Forms Logarithm Form 5 = log (3) Logs Give you Eponents Input Becomes Output 5 = 3 Base Stays the Base Eponential Form
Eamples Write each equation in eponential form 1.log 15 (5) = /3.Log 8 () = 1/3 15 /3 = 5 8 1/3 = Write each equation in logarithmic form 1.If 64 = 4 3.If 1/7 = 3 log 4 (64) = 3 Log 3 (1/7) =
Eample Complete the table if a is a positive real number and: f a Domain Range Continuous? One-to-One? Concavity Left End Behavior Right End Behavior 1 f a f All Reals All Positive Reals Yes Yes Always Up lim a 0 lim a log a All Positive Reals All Reals Yes Yes Always Down 0 lim log a lim log a
The Change of Base Formula The following formula allows you to evaluate any valid logarithm statement: log b a log log c c a b For a and b greater than 0 AND b 1. Eample: Evaluate log1.04 ln ln 1.04 17.673
Solve: Solving Equations with the Change of Base Formula 3.46 1909 Isolate the base and power Change the eponential equation to an logarithm equation Use the Change of Base Formula 3.46 908 3.46 454 log 454 3.46 log 454 log 3.46 4.989
Properties of Logarithms For a>0, b>0, m>0, m 1, and any real number n. Logarithm of 1: log 1 0 m Logarithm of the base: log 1 m m Power Property: log m a n nlog a m Product Property: ab a b log log log m m m Quotient Property: a log log a log b m b m m
Eample 1 Condense the epression: 1 3 log log 10 log 7 5 5 5 13 log log 10 log 7 5 5 5 3 log 10 log 7 5 5 log 10 3 5 7
Eample Epand the epression: ln 3y ln 3 ln ln y ln 3 ln ln y
Eample 3 Solve the equation: 4 4 log 3 4 4 3 log log 3 3
AP Reminders Do not forget the following relationships: ln e e ln e a e b e a b e e a b e ab
Inverse Trigonometry
Tangent Cosine Sine Trigonometric Functions Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore, there inverses are not functions. Cosecant Secant Cotangent
Tangent Cosine Sine Trigonometric Functions with Restricted Domains D :, D : 0, D :, In order for their inverses to be functions, the domains of the trigonometric functions are restricted so that they become oneto-one. D :,0 0, D : 0,, D : 0, Cosecant Secant Cotangent
Trigonometric Functions with Restricted Domains Function Domain Range f () = sin f () = cos f () = tan f () = csc f () = sec f () = cot, 0,,,0 0, 0,, 0, 1,1 1,1,,, 1 1,, 1 1,
Tan -1 Cos -1 Sin -1 Inverse Trigonometric Functions Csc -1 Sec -1 Cot -1
Inverse Trigonometric Functions Function Domain Range f () = sin -1 f () = cos -1 f () = tan -1 f () = csc -1 f () = sec -1 f () = cot -1 1,1 1,1,,, 1 1,, 1 1,, 0,,,0 0, 0,, 0,
Alternate Names/Defintions for Inverse Trigonometric Functions Familiar Alternate Calculator f () = sin -1 f () = arcsin f () = sin -1 f () = cos -1 f () = arccos f () = tan -1 f () = arctan f () = cos -1 f () = tan -1 f () = csc -1 f () = arccsc f () = sin -1 1/ f () = sec -1 f () = arcsec f () = cos -1 1/ f () = cot -1 f () = arccot f () = -tan -1 + Arccot is different because it is always positive but tan can be negative.
Evaluate: Eample 1 sin 1 1 This epression asks us to find the angle whose sine is ½. Remember the range of the inverse of sine is,. 1 Since sin and 6, 6 sin 6 1 1
Eample Evaluate: csc 1 1 This epression asks us to find the angle whose cosecant is -1 (or sine is -1). Remember the range of the inverse of cosecant is,0 0,. Since csc 1 and 0, 1 csc 1
Eample 3 Evaluate: tan arcsin 1 3 The embedded epression asks us to find the angle whose sine is 1/3. Draw a picture (There are infinite varieties): 3 a Find the missing side length(s) a 1 3 1 a 8 It does not even matter what the angle is, we only need to find: tan opp adj 1 4 Is the result positive or negative? Since arcsin 0, 1 3 tan 0
Eample 4 Evaluate: tan cos 1 ( 1 ) 6 The embedded epression asks us to find the angle whose cosine is -1/6. Draw a picture (There are infinite varieties): Ignore the negative for now. 6 1 Find the missing side length(s) o 1 6 o 35 o tan It does not even matter what the angle is, we only need to find: opp adj 35 1 35 Is the result positive or negative? 1 1 Since cos, tan 0 6
Eample 3 Evaluate: costan 1 The embedded epression asks us to find the angle whose tangent is. Draw a possible picture (There are infinite varieties): h 1 Find the missing side length(s) 1 h h 1 It does not even matter what the angle is, we only need to find: cos adj hyp 1 1 Is the result positive or negative? Since - tan, 1 cos 0
White Board Challenge Evaluate without a calculator: sec 1 4
White Board Challenge Evaluate without a calculator: cot 3 1 3 3
White Board Challenge Evaluate without a calculator: arccos 3
White Board Challenge Evaluate without a calculator: cot csc 1 5 6
White Board Challenge Evaluate without a calculator: tan sin 1 1