1 / 14

Understanding Functions, Inverses, and Logarithms

Discover the concepts of one-to-one functions, inverses, and logarithms with detailed examples and properties. Learn how to find inverses, solve logarithmic equations, and use logarithmic properties. Calculate compound interest using logarithms.

buffy
Télécharger la présentation

Understanding Functions, Inverses, 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. Functions and Logarithms

  2. One-to-One Functions • A function f(x) is one-to-one if f(a) ≠ f(b) whenever a ≠ b. • Must pass horizontal line test. Not one-to-one One-to-one

  3. Inverses • If a function is one-to-one, then it has an inverse. • Notation: f-1(x) • IMPORTANT:

  4. Inverses • To find the inverse of a function, solve for x, “swap” x and y, and solve for y. • Ex: What is the inverse of f(x) = -2x + 4? y = -2x + 4 (“swap” x and y) y – 4 = -2x Therefore,

  5. Inverses • Inverse functions are symmetric (reflected) about the line y = x. • Therefore, in order for two functions to be inverses, the results of the composites is x. AND

  6. Inverses • Example: Prove the two functions from our last example are inverses. Now, we must check the other composite! = x Therefore, these two functions must be inverses of one another!!! = x

  7. Logarithms • y = logax means ay = x • Ex: log381 = 4 means 34 = 81 • What is the inverse of y = logax? • Since y = logax is ay = x, then the inverse has to be • ax = y

  8. Logarithms • Common logarithms: log x means log10x ln x means logex

  9. Inverse Properties of Logs (both of these hold true if a > 1 and x > 0) (both of these hold true if x > 0)

  10. Inverse Properties of Logs • Example: Solve ln x = 3t + 5 for x. (use each side as an exponent of e) (e and ln are inverses and “undo” each other.)

  11. Inverse Property of Logs • Example: Solve e2x = 10 for x. (take the natural log of both sides) (ln and e are inverses and “undo” each other.)

  12. Properties of Logarithms • For any real numbers x > 0 and y > 0,

  13. Change of Base Property • Since our calculators will not calculate logs of bases other than 10 or e,

  14. Example • Sarah invests $1000 in an account that earns 5.25% interest compounded annually. How long will it take the account to reach $2500? (divide both sides by 1000) (take a log of both sides…doesn’t matter what base you use!!!) (by my property, exponent comes out front) t ≈ 17.907 years (divide by ln(1.0525))

More Related