Section 7.1: Inverse Functions
Definition of a One-to-One Function
A function f is one-to-one if no two input values yield the same output. That is, for
all x, y in the domain of f we have
Checking Whether a Function is One-to-One
Horizontal Line Test: If no horizontal line intersects the graph of a function in more
than one point, then the function is one-to-one.
A function that is always increasing or decreasing is one-to-one. This happens when
the derivative of a function is always positive or negative.
Definition of Inverse Function:
If f is a one-to-one function with domain A and range B, then the inverse of f,
denoted f −1, is a function with domain B and range A defined by
Finding the inverse of a function:
1. Write y = f (x)
2. Solve for x in terms of y.
3. Switch the x and y in the resulting equation to obtain the inverse.
Finding (f −1)'(a) without finding f −1:
Use the formula
Note: You must be able to find f −1(a), usually by inspection.
Section 7.2: Exponential Functions
Definition of Exponential Function:
An exponential function is of the form
where a is a positive constant.
Properties of Exponential Functions:
If a > 0 and a ≠ 1, then is a continuous function with domain R and range
(0,∞). If a, b > 0 then
Limits of Exponential Functions:
If a > 1, then and .
If 0 < a < 1, then and
Definition of the Number e:
e is the number such that
Derivative of the Natural Exponential Function:
The derivative of ex is simply itself:
Using the chain rule, we get:
Use the formula:
Section 7.3: Logarithmic Functions
Definition of Logarithmic Function:
The inverse function of the exponential function is called the logarithmic
function with base a and is written . The logarithmic function is defined by
Cancellation Laws for Logarithms:
The following cancellation equations hold for logarithms:
Properties of Logarithms:
If a > 1, the function is a one-to-one, continuous, and increasing
function with domain (0,∞) and range R. If x, y > 0 and r is a real number then
the following properties hold:
Limits of Logarithms:
If a > 1, then and
Definition of Natural Logarithm:
The logarithm with base e is called the natural logarithm and is written
Properties of Natural Logarithm:
The following are defining properties of the natural logarithm
Change of Base Formula:
For any positive number a (a ≠ 1), we have
This formula allows us to use a calculator to approximate logarithms of any base.
Section 7.4: Derivatives of Logarithmic Functions
Derivative of the Natural Logarithm
Use the formula
Applying the chain rule gives the formula
where u is a function in terms of x.
Integrate by means of the formula
Integration of Tangent
By finding tan x d x using the integral of 1/x formula, we obtain the following result
Differentiation Formula for Logarithmic Functions
Differentiation Formula for Exponential Functions
Integration Formula for Exponential Functions
Logarithmic differentiation is useful for taking the derivatives of complicated functions
containing products, quotients, or exponents.
Procedure for Logarithmic Differentiation:
1. Take the natural logarithm of both sides of an equation y = f (x) and then
simplify using the properties of logarithms.
2. Differentiate both sides of the equation with respect to x being sure to apply
the chain rule where necessary.
3. Solve the resulting equation for y'.