3.2 Inverse Functions and Logarithms 3.3 Derivatives of Logarithmic and Exponential functions

One-to-one functions Definition: A function f is called a one-to-one function if it never takes on the same value twice; that is f(x 1 ) ≠ f(x 2 ) whenever x 1 ≠ x 2. Horizontal line test: A function f is one-to-one if and only if no horizontal line intersects its graph more than once. Examples: f(x) = x 3 is one-to-one but f(x) = x 2 is not.

Inverse functions Definition: Let f be a one-to-one function with domain A and range B. Then the inverse function f -1 has domain B and range A and is defined by for any y in B. Note: f -1 (x) does not mean 1 / f(x). Example: The inverse of f(x) = x 3 is f -1 (x)=x 1/3 Cancellation equations:

How to find the inverse function of a one-to-one function f Step 1: Write y=f(x) Step 2: Solve this equation for x in terms of y (if possible) Step 3: To express f -1 as a function of x, interchange x and y. The resulting equation is y = f -1 (x) Example: Find the inverse of f(x) = 5 - x 3

Another example: Switch x and y : Inverse functions are reflections about y = x. Solve for x :

At x = 2 : At x = 4 : Slopes are reciprocals. First consider an example: Derivative of inverse function

Calculus of inverse functions Theorem: If f is a one-to-one continuous function defined on an interval then its inverse function f -1 is also continuous. Theorem: If f is a one-to-one differentiable function with inverse function f -1 and f ′ (f -1 (a)) ≠ 0, then the inverse function is differentiable and Example: Find (f -1 ) ′ (1) for f(x) = x 3 + x + 1 Solution: By inspection f(0)=1, thus f -1 (1) = 0 Then

Consider where a>0 and a≠1 This is a one-to-one function, therefore it has an inverse. The inverse is called the logarithmic function with base a. Example: The most commonly used bases for logs are 10: and e : is called the natural logarithm function. Logarithmic Functions

Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

Derivatives of Logarithmic and Exponential functions Examples on the board.

Logarithmic Differentiation The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms. Step 1: Take natural logarithms of both sides of an equation y = f (x) and use the properties of logarithms to simplify. Step 2: Differentiate implicitly with respect to x Step 3: Solve the resulting equation for y ′ Examples on the board