**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 e^{x }is simply itself:

Using the chain rule, we get:

**Integrating e ^{x}**

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.

**Integration of **

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**

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'.