Inverse functions & Logarithms

P.4

Vocabulary

One-to-One Function: a function f(x) is one-to-one on a domain D if f(a) ≠ f(b) whenever a ≠ b. The graph of a one-to-one function y = f(x) can intersect any horizontal line at most once. (The Horizontal Line)

Inverse of f: the function defined by reversing a one-to-one function f. The symbol for the inverse is f-1.

Identity function: the result of composing a function and its inverse in either order.

A test for Inverses

Functions f and g are an inverse pair if and only if f(g(x)) = x and g(f(x)) = x. In this case, g = f-1 and f = g-1.

Example 1: Testing for Inverses

A) f(x) = x2 and g(x)

B) f(x) = x + 1 g(x) = x - 1

2

1

x

Finding Inverses

The domain of f-1 is the range of f. The range of f-1 is the domain of f. To draw the graph of f-1, reflect the system in the line y = x.

Example 2: Finding the Inverse Function 2-Ways

A) y = B) y =

23

1x 2

3

1x

Logarithmic functions

The base a logarithm function y = loga x is the inverse of the base a exponential function y = ax ( a > 0, a ≠ 1). The domain of loga x is ( 0, ∞), the range of y = ax. The range of loga x is (-∞, ∞), the domain of ax.

Properties of logarithms

Inverse Properties of ax and loga x

•1) Base a : = x and loga ax = x a>0, a≠1, x>0

•2) Base e: e lnx = x and ln ex = x x>0

Arithmetic Properties ( x>0 and y>0)

•1) Product Rule: loga xy = loga x + loga y

•2) Quotient Rule: loga = loga x - loga y

•3) Power Rule: loga xy = y∙loga x

xa xa log

y

x

Example 3: Using Logarithms

A) Solve for x: ln x = 3t - 5

B) Write as power of e: 5-3x

Change of base formula

Every logarithmic function is a constant multiple of the natural logarithm.

loga x = a >0, a ≠ 1

We need the change of base formula to use our calculators. Our calculators only understand ln or log10 (log).

a

x

ln

ln

Example 4: Finding Time

Mary invests $500 in an account that earns 3% interest compounded annually. How long will it take the account to reach $1250?