Inverse functions and logarithms

Transcription

1 Inverse unctions and logarithms Recall that a unction is a machine that takes a number rom one set and puts a number o another set. Must be welldeined, meaning the unction is decisive: () always has an answer and () always puts out one answer or each number taken in. Examples:. : R! R deined by x 7! x ; e.g. /3 etc. 4 /9. : R 0! R 0 deined by x 7! p x ; e.g. 4 /9 etc. /3 Note that p x is only a unction when we go to extra e ort to decide that we re always going to choose the positive answer.

2 3. Let bacteria grow, and measure population over time. Consider N : N! N by N(t) =#bacteriaattimet. t N(t) = pop. at time t (hours) Now suppose we we re trying to ask the question how long will it take to grow at least 500 bacteria? Answer: between 4 and 5 hours Inverse unctions Given a unction,theinverseunction is the machine that takes in s output, and returns the corresponding input. In notation, we write that x 7! (x) 7! x ( (x)) = x and ( (x)) = x. Example: I : R 0! R 0 is given by (x) =x, then (x) = p x = p x. NonExample: I : R! R 0 is given by (x) =x, then (x) isnot welldeined.

3 I : R 0! R 0 is given by (x) =x, then (x) = p x I y = x and x 0, then x = p y. I : R! R 0 is given by (x) =x, then p x is not the inverse I y = x and x < 0, then x 6= p y!

4 When is a unction invertible? A unction is onetoone i no two inputs give the same output, that is, i x 6= x,then (x ) 6= (x ). Example: over all real numbers, (x) =x is not onetoone. However, over nonnegative real numbers, (x) =x is onetoone no! yes! Horizontal line test: A unction is onetoone i and only i no horizontal line intersects the unction s graph more than once. Answer: A unction is invertible i and only i it is onetoone. Graphing inverses For a onetoone unction,wehave (x) =y i and only i x = (y) The graph o y = (x) is the relection o the graph o over the line y = x (i.e. swap the axes). Further, the domain o is the range o, and the range o is the domain o.

5 You try: I For each o the ollowing unctions, (a) give the domain and range o,and(b) decide i is invertible. I I is invertible, then (c) sketch a graph o, (d) give the domain and range o,and(e) try to write a ormula or. I I is not invertible over all o the real numbers, what is a restricted domain over which is invertible? Over that restricted domain, do (c) and (d) rom above. () (x) = x () (x) =x 5 (3) (x) = cos(x) (4) (x) =/(x + )

6 Calculating the inverse unction algebraically Given an invertible, solve or by setting (y) =x, and solving or y = (x). Example: Let (x) =/(x + ). Set x = (y) =/(y + ). Then y +=/x, so that (x) =y =(/x). Example: Set Then Let (x) =x 3 +. (Check: is it invertible??) x = (y) =y 3 +. y 3 = x, so that (x) =y =(x ) /3. Checking your answer algebraically Recall that is deined by ( (x)) = x and ( (x)) = x. Example: We calculated that i (x) =/(x + ), then (x) =(/x). Let s check! ( (x)) = /((/x) + ) =/(/x) =x X and ( (x)) = (//(x + )) = x + =x X You try:. Check that i (x) =x 3 +then (x) =(x ) /3 by calculating ( (x)) and ( (x)).. For the ollowing unctions, calculate (x) and veriy your answer as above. (a) (x) =3/(x ) (b) (x) =5 p x

7 Logarithms The exponential unction a x has inverse log a (x), i.e. log a (a x )=x = a log a (x), i.e. y = a x i and only i log a (y) =x. y=a x y=log a (x) Properties o Logarithms a=. a=0 y=a x y=log a (x) Domain: (0, ) i.e.allx > 0 Range: (, ) i.e.allx

8 Properties o Logarithms 0 < a < : y=a x a=0. y=log a (x) a=0.8 a=0.8 a=0. Domain: (0, ) i.e.allx > 0 Range: (, ) i.e.allx Properties o Logarithms Since... we know.... a 0 =. a = a 3. a b a c = a b+c 4. (a b ) c = a b c. log a () = 0. log a (a) = 3. log a (b c) = log a (b) + log a (c) 4. log a (b c )=c log a (b) Example: why log a (b c) = log a (b) + log a (c): Suppose y = log a (b) + log a (c). Then a y = a log a(b)+log a (c) = a log a(b) a log a(c) = b c. So y = log a (b c) aswell! Lastly: log a (b) log a (c) = log c(b)

9 Favorite logarithmic unction Remember: y = e x is the unction whose slope through the point (0,) is. The inverse to y = e x is the natural log: ln(x) = log e (x) y=e x m= y=ln(x) m= We will oten use the acts that e ln(x) = x (or x > 0) and ln(e x )=x (or all x) Two super useul acts: Explain why: () log a (b) =ln(b)/ ln(a) () a b = e b ln(a) [hint: start by rewriting b ln(a), and use the act that e ln(x) = x]

10 Examples: () Condense the logarithmic expressions ln(x)+3 ln(x+) ln(x+5) ln(x) 3 (log 3(x) log 3 (x+)) () Solve the ollowing expressions or x: e x = e 3x 4 3( x ) = 4 (e 3x 5 ) 5 = ln(3x + ) ln(5) = ln(x) Inverse trig unctions Two notations: (x) (x) sin(x) sin (x) =arcsin(x) cos(x) cos (x) = arccos(x) tan(x) tan (x) =arctan(x) sec(x) sec (x) =arcsec(x) csc(x) csc (x) =arccsc(x) cot(x) cot (x) = arccot(x) There are lots o points we know on these unctions... Examples:. Since sin( /) =, we have arcsin() = /. Since cos( /) = 0, we have arccos(0) = / Etc...

11 In general: arc ( ) takes in a ratio and spits out an angle: θ! " # cos( ) =a/c so arccos(a/c) = sin( ) =b/c so arcsin(b/c) = tan( ) =b/a so arctan(b/a) = Domain problems: sin(0) = 0, sin( ) = 0, sin( ) = 0, sin(3 ) = 0,... So which is the right answer to arcsin(0), really? Graphs arcsin(x) arccos(x) arctan(x)!!/!/ "!/ 0 "!/ arcsec(x) arccsc(x) arccot(x)!!!/ 0 "!/ 0