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) = x 2 and g(x) B) f(x) = x + 1 g(x) = x - 1

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 =

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

Properties of logarithms Inverse Properties of a x and log a x 1) Base a : = x and log a a x = x a>0, a≠1, x>0 2) Base e: e lnx = x and ln e x = x x>0 Arithmetic Properties ( x>0 and y>0) 1) Product Rule: log a xy = log a x + log a y 2) Quotient Rule: log a = log a x - log a y 3) Power Rule: log a x y = y∙log a 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. log a x =a >0, a ≠ 1 We need the change of base formula to use our calculators. Our calculators only understand ln or log 10 (log).

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?