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Understanding Inverse and Exponential Functions through Real-World Scenarios

This educational piece delves into the concepts of inverse and exponential functions using relatable driving and growth scenarios. It explains how to determine distance from home based on time traveled and vice versa, using equations to illustrate these relationships. The article also discusses exponential growth in financial terms, demonstrating how values can double over time. Through practical examples and mathematical notations, readers will grasp these essential concepts and their applications in real life.

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Understanding Inverse and Exponential Functions through Real-World Scenarios

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  1. Two really important ideas Function Inverse & Exponential Function

  2. Function Inverse

  3. 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?

  4. 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? d=number of miles away from my house t=number of hours I’ve been driving d=30t+10

  5. Going Driving I 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 house t=number of hours I’ve been driving d=30t+10

  6. Going Driving I 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 house t=number of hours I’ve been driving d=30t+10 (d-10)/30=t

  7. Going driving

  8. I haven’t changed anything(except my point of view) (d-10)/30=t d=30t+10

  9. I haven’t changed anything(except my point of view) 40 miles 1 hours 40 miles 1 hours 10 miles 0 hours 0 hours 10 miles (d-10)/30=t d=30t+10

  10. Cubing I have an equation y=x3. I know x=2 and I want to figure out y. y=(2)3 y=8 I have an equation y=x3. I know y=8 and I want to figure out x. 8=(x)3 y=∛8=2

  11. Cubing to cube root

  12. Cubing to cube root y=x3 x=∛y

  13. Cubing to cube root The relationship between x and y stays the same Only my point of view changes y=x3 x=∛y

  14. Notation From x  y y=x3 ƒ(x)=x3

  15. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y

  16. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y

  17. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y ƒ-1(x)=∛x Because x and y don’t actually mean anything, I can change their names if I want.

  18. Notation From x  y y=x3 ƒ(x)=x3 From y  x y=x3 ∛y=x ∛y=ƒ-1(y) ƒ-1(y)=∛y ƒ-1(x)=∛x Because x and y don’t actually mean anything, I can change their names if I want. This is not actually a good idea, but it’s popular in many math books

  19. How to find a function inverse • ƒ(x)=………….x…………. • Rewrite as y=……………x………… • Solve for y. x=~~~~y~~~~~~ • Rewrite as an inverse ƒ-1(y)=~~~~y~~~~~~ • OPTIONAL: change ys to xs. • ƒ-1(x)=~~~~x~~~~~~ • WARNING: Always check that your inverse is actually a function.

  20. Round trip I 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 home x=number of hours since I started driving

  21. Round Trip

  22. 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 can’t figure out y (time). x is NOT a function of y.

  23. Testing if the inverse is a function

  24. 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 + 34.5 • t-1(x) = 34.5 + x • t-1(x) = x – 1/34.5 • None of the above.

  25. 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-34.5 t-1(x)=x-34.5 E

  26. Exponential Functions The “I’m going to lie to you a lot” version

  27. Exponential functions measure steady growth • If you really want to know what that means exactly, take differential equations (after Calculus) • Here’s the basic (lying) version • An exponential growth happens when something is making more of itself (in a “steady” way) • People, money, bacteria, etc…

  28. Example • One dollar makes one dollar every year. $1 $1 Year 0 Year 1

  29. Example • One dollar makes one dollar every year. $1 $1 $1 Year 0 Year 1

  30. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  31. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2

  32. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  33. Example • One dollar makes one dollar every year. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  34. Example • Every year I keep what I have and add what I have. $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  35. Example • Every year I double my money $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  36. Example • Every year I double my money $1 y=1(2x) y=# of $ x=# of yrs $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 $1 Year 0 Year 1 Year 2 Year 3

  37. Exponential Growth

  38. Exponential Decay

  39. 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 above

  40. 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 above

  41. Anatomy • The standard form of the exponential is y=abx • a is called the initial value (y-intercept) • b is called the growth factor. • When 0<b<1, you have exponential decay • A non-standard form is y=ac-x • a is the initial value (y-intercept) • c-1=1/c is the growth factor. • When 0<1/c<1, you have exponential decay

  42. The compound interest formula • P dollars are invested at r% per year compounded n times per year. After t years, I have A dollars.

  43. The compound interest example • 100dollarsare invested at 7% per year compounded monthly. How many dollars do I have after 5 years? • P=100,r=7/100,n=12 (12 months a year),t=5 Find A.

  44. The compound interest example • I invested some money at 3% per year compounded quarterly. After 9 yearsI had $1000. How much did I start with? • R=3/100, n=4,t=9,A=1000,find P.

  45. Asymptotes • All exponentials y=abx have asymptote y=0

  46. Asymptotes • Adding c moves a graph up by c.

  47. Asymptotes • Any function y=abx+c has asymptote y=c

  48. Consider the function below: Which of the following statements matches with this function? As x approaches infinity, f(x) approaches 0. As x approaches negative infinity, f(x) approaches 0. As x approaches infinity, f(x) approaches -4. As x approaches negative infinity, f(x) approaches -4. None of the above

  49. D) As x approaches negative infinity, f(x) approaches -4.

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