Chapter 1.5Functions 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

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

Horizontal Line Test Examples X3 x2

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)

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

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 (-,∞, ,∞)

Important Log Functions Two very important logs for conversions and our calculators are:

The common log function Log10x = logx

The natural log Logex = lnx

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

Examples: Solve for x 1) lnx = 3y + 5

2) e2x = 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)

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

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