1 / 17

1.5 Functions and Logarithms

Learn about one-to-one functions, inverse functions, and logarithms. Discover the properties of logarithms and how to use them in various calculations and regression equations.

falkner
Download Presentation

1.5 Functions and Logarithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 1.5 Functions and Logarithms Golden Gate Bridge San Francisco, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington

  2. In other words, a function is one-to-one on domain D if: whenever A relation is a function if: for each x there is one and only one y. A relation is a one-to-one if also: for each y there is one and only one x.

  3. To be one-to-one, a function must pass the horizontal line test as well as the vertical line test. one-to-one not one-to-one not a function (also not one-to-one)

  4. Inverse functions: Given an x value, we can find a y value. Solve for x: Inverse functions are reflections about y = x. Switch x and y: (eff inverse of x)

  5. WINDOW GRAPH a parametrically: Y= example 3: Graph: for

  6. b Find the inverse function: WINDOW Y= > GRAPH example 3: Graph: for Switch x & y: Change the graphing mode to function.

  7. is called the natural log function. is called the common log function. Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Two raised to what power is 16? Example: The most commonly used bases for logs are 10: and e:

  8. In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work. is called the natural log function. is called the common log function.

  9. Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

  10. Example 6: $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent. 17.9 years In real life you would have to wait 18 years.

  11. Indonesian Oil Production (million barrels per year): Example 7: Use the natural logarithm regression equation to estimate oil production in 1982 and 2000. How do we know that a logarithmic equation is appropriate? In real life, we would need more points or past experience.

  12. ENTER ENTER STO Indonesian Oil Production: 20.56 million 42.10 70.10 60 70 90 60,70,90 2nd { 2nd } alpha L 1 6 3 5 alpha L 1 , alpha L 2 2nd MATH LnReg The calculator should return: Statistics Regressions Done

  13. ENTER ENTER 6 3 5 alpha L 1 , alpha L 2 2nd MATH LnReg The calculator should return: Statistics Regressions Done 6 8 2nd MATH Statistics ShowStat The calculator gives you an equation and constants:

  14. WINDOW ENTER ENTER Y= We can use the calculator to plot the new curve along with the original points: x y1=regeq(x) ) regeq 2nd VAR-LINK Plot 1

  15. WINDOW GRAPH ENTER ENTER Plot 1

  16. WINDOW GRAPH

  17. F3 ENTER ENTER Trace What does this equation predict for oil production in 1982 and 2000? This lets us see values for the distinct points. This lets us trace along the line. 82 Enters an x-value of 82. Moves to the line. In 1982, production was 59 million barrels. 100 Enters an x-value of 100. p In 2000, production was 84 million barrels.

More Related