1. Chapter 1.5 Functions 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 X 3 x 2

4. Finding the inverse function If a function is one-to-one it has an inverse Writing f -1 as 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 = log a x is the inverse of the base a exponential function y = a x The domain of log a x is (0,∞). The range of log a x is (-,∞,,∞)

7. Important Log Functions Two very important logs for conversions and our calculators are: The common log function Log 10 x = logx The natural log Log e x = lnx

8. Properties of Logarithms Inverse properties for a x and log a x 1) Base a: a loga(x) = x, log a a x = x, a > 1, x > 0 2) Base e: e lnx = x, lne x = x

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

10. Properties of Logarithms For any real number x > 0 and y > 0 1) Product Rule: log a xy = log a x + log a y 2) Quotient Rule: log a (x/y) = log a x – log a y 3) Power Rule: log a x y = ylog a x 4) Change of Base Formula: log a x = (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