two really important ideas

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Two really important ideas Function Inverse & Exponential Function

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Two really important ideas. Function Inverse & Exponential Function. Function Inverse. Going Driving. I start 10 miles away from my house and drive away from my house at 30 mph. If I know how long I’ve been driving, how far am I from my house?. Going Driving. - PowerPoint PPT Presentation

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A jewelry maker has total revenue for her necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces produces and sold. How many necklaces must be produced and sold in order to break even?

Two really important ideasFunction Inverse&Exponential FunctionFunction InverseGoing DrivingI start 10 miles away from my house and drive away from my house at 30 mph. If I know how long Ive been driving, how far am I from my house?Going DrivingI start 10 miles away from my house and drive away from my house at 30 mph. If I know how long Ive been driving, how far am I from my house?d=number of miles away from my houset=number of hours Ive been drivingd=30t+10Going DrivingI start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving?d=number of miles away from my houset=number of hours Ive been drivingd=30t+10Going DrivingI start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving?d=number of miles away from my houset=number of hours Ive been drivingd=30t+10(d-10)/30=t

Going driving

I havent changed anything(except my point of view)

d=30t+10(d-10)/30=tI havent changed anything(except my point of view)

10 miles10 miles0 hours0 hours40 miles1 hours1 hours40 milesd=30t+10(d-10)/30=tCubingI have an equation y=x3. I know x=2 and I want to figure out y.y=(2)3y=8I have an equation y=x3. I know y=8 and I want to figure out x.8=(x)3y=8=2

Cubing to cube root

Cubing to cube root

y=x3x=yCubing to cube root

y=x3x=yThe relationship between x and y stays the sameOnly my point of view changesNotationFrom x yy=x3(x)=x3NotationFrom x yy=x3(x)=x3From y xy=x3y=xy=-1(y)-1(y)=y

NotationFrom x yy=x3(x)=x3From y xy=x3y=xy=-1(y)-1(y)=y

NotationFrom x yy=x3(x)=x3From y xy=x3y=xy=-1(y)-1(y)=y-1(x)=xBecause x and y dont actually mean anything, I can change their names if I want.NotationFrom x yy=x3(x)=x3From y xy=x3y=xy=-1(y)-1(y)=y-1(x)=xBecause x and y dont actually mean anything, I can change their names if I want.This is not actually a good idea, but its popular in many math booksHow to find a function inverse(x)=.x.Rewrite as y=xSolve for y. x=~~~~y~~~~~~Rewrite as an inverse -1(y)=~~~~y~~~~~~OPTIONAL: change ys to xs. -1(x)=~~~~x~~~~~~WARNING: Always check that your inverse is actually a function.

Round tripI drive away from home for 1.25 hours at 30 miles per hour, then I turn around and drive back home at 30 miles per hour.y=number of miles I am from homex=number of hours since I started drivingRound Trip

Round Trip

If I know x (time), I can figure out y (distance).y is a function of x.If I know y (distance),I cant figure out y (time). x is NOT a function of y.Testing if the inverse is a function

A shoe size that is size x in the United States is size t(x) in Continental size, where t(x)=x+34.5 Find a function that will convert Continental shoe size to a US shoe size. t-1(x) = 1/(x+34.5) t-1(x) = 1/x + 34.5t-1(x) = 34.5 + xt-1(x) = x 1/34.5 None of the above. A shoe size that is size x in the United States is size t(x) in Continental size, where t(x)=x+34.5 Find a function that will convert Continental shoe size to a US shoe size. t(x)=x+34.5 y=x+34.5 y-34.5=x y-34.5=t-1(y)t-1(y)=y-34.5t-1(x)=x-34.5 E

Exponential FunctionsThe Im going to lie to you a lot versionExponential functions measure steady growthIf you really want to know what that means exactly, take differential equations (after Calculus)Heres the basic (lying) versionAn exponential growth happens when something is making more of itself (in a steady way)People, money, bacteria, etcExampleOne dollar makes one dollar every year.$1Year 0$1Year 1ExampleOne dollar makes one dollar every year.$1Year 0$1Year 1$1ExampleOne dollar makes one dollar every year.$1Year 0$1Year 1$1$1Year 2$1ExampleOne dollar makes one dollar every year.$1Year 0$1Year 1$1$1Year 2$1$1$1ExampleOne dollar makes one dollar every year.$1Year 0$1Year 1$1$1Year 2$1$1$1Year 3$1$1$1$1ExampleOne dollar makes one dollar every year.$1Year 0$1Year 1$1$1Year 2$1$1$1Year 3$1$1$1$1$1$1$1$1ExampleEvery year I keep what I have and add what I have.$1Year 0$1Year 1$1$1Year 2$1$1$1Year 3$1$1$1$1$1$1$1$1ExampleEvery year I double my money$1Year 0$1Year 1$1$1Year 2$1$1$1Year 3$1$1$1$1$1$1$1$1ExampleEvery year I double my money$1Year 0$1Year 1$1$1Year 2$1$1$1Year 3$1$1$1$1$1$1$1$1y=1(2x)y=# of $x=# of yrs

Exponential Growth

Exponential Decay

Which of the following functions represent that of exponential decay?f(x)=(1/2)x f(x)=(1/2)-x f(x)=(1/3)-x (b) and (c)None of the aboveWhich of the following functions represent that of exponential decay?f(x)=(1/2)x f(x)=(1/2)-x f(x)=(1/3)-x (b) and (c)None of the above

AnatomyThe standard form of the exponential is y=abxa is called the initial value (y-intercept)b is called the growth factor.When 0