1 / 12

Chapter 1.5 Functions and Logarithms

Chapter 1.5 Functions and Logarithms. One-to-One Function. A function f(x) is one-to-one on a domain D (x-axis) if f(a) ≠ f(b) whenever a≠b Use the Horizontal line test

caldwell-morton
Download Presentation

Chapter 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. Chapter 1.5Functions and Logarithms

  2. One-to-One Function • A function f(x) is one-to-one on a domain D (x-axis) if f(a) ≠ f(b) whenever a≠b • Use the Horizontal line test • The graph of a one-to-one function y = f(x) can intersect any horizontal line at most once. If a horizontal intersects a graph more than once, the function is not one. • If a function is one-to-one it has an inverse

  3. Horizontal Line Test Examples • X3x2

  4. Finding the inverse function • If a function is one-to-one it has an inverse • Writing f-1as a Function of x • 1) Solve the equation y = f(x) for x in terms of y. • 2) Interchange x and y. The resulting formula will be y = f-1 (x)

  5. Inverse Examples • Show that the function y = f(x) = -2x +4 is one-to-one and find its inverse • Every horizontal line intersects the graph of f exactly once, so f is one-to-one and has an inverse • Step 1: Solve for x in terms of Y: • Y = -2x + 4 • X= -(1/2)y +2 • Step 2: Interchange x and y: y = -(1/2)x + 2

  6. Logarithmic Functions • The base a logarithm function y = logax is the inverse of the base a exponential function y = ax • The domain of logax is (0,∞). The range of logax is (-,∞, ,∞)

  7. Important Log Functions • Two very important logs for conversions and our calculators are: • The common log function • Log10x = logx • The natural log • Logex = lnx

  8. Properties of Logarithms • Inverse properties for ax and logax • 1) Base a: aloga(x) = x, logaax = x, a > 1, x > 0 • 2) Base e: elnx = x, lnex = x

  9. Examples: Solve for x • 1) lnx = 3y + 5 • 2) e2x = 10

  10. Properties of Logarithms • For any real number x > 0 and y > 0 • 1) Product Rule: logaxy = logax + logay • 2) Quotient Rule: loga(x/y) = logax – logay • 3) Power Rule: logaxy = ylogax • 4) Change of Base Formula: logax = (lnx)/(lna)

  11. Investment • Sarah invests $1000 in an account that earns 5.25% interest compounded annually. How long will it take the account to reach $2500? • P(1+(r/c))ct=A • 1000(1.0525)t = 2500 • (1.0525)t = 2.5 • Ln(1.0525)t = ln2.5 • Tln1.0525 = ln2.5 • T = (ln2.5)/(ln1.0525) = 17.9

  12. Homework • Quick Review: pg 43, # 1, 3, 7, 9 • Exercises: pg 44, # 1, 2, 3, 6, 7, 8, 10, 33, 34, 37, 39, 40, 47, 48

More Related