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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

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Chapter 1.5 Functions and Logarithms

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  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

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