**Domain and Range of **

Inverse Functions

•If f is one-to-one, its inverse is a

function.

•The domain of f ^{-1 }is the range of f.

•The range of f ^{-1 }is the domain of f

•Example

**Problem: **Verify that the inverse of

** Graphs of Inverse Functions**

•The graph of a function f and

its inverse f ^{-1}are symmetric with respect

to the line y = x.

•Example.

**Problem:** Find the graph of the inverse

function

**Answer:**

**Finding Inverse Functions**

•If y= f(x),

•Inverse if given implicitly by x= f(y).

• Solve for y if possible to get y= f ^{-1}(x)

•Process

•Step 1: Inter change x and y to obtain an

equation x= f(y)

•Step 2: If possible, solve for yin terms of

x.

•Step 3: Check the result.

•Example.

**Problem: **Find the inverse of the function

**Answer :**

**Restricting the Domain**

•If a function is not one-to-one, we can

often restrict its domain so that the

new function is one-to-one.

•Example.

**Problem:** Find the inverse of

if the domain of f is x≥0.

**Answer:**

**Key Points**

•One-to- One Functions

•Horizontal-line Test

•Inverse Functions

•Domain and Range of Inverse Functions

•Graphs of Inverse Functions

•Finding Inverse Functions

•Restricting the Domain

**Exponential **

Functions

Section 4.3

**Exponents**

•For negative exponents :

•For fractional exponents :

•Example.

**Problem: **Approximate 3πto five decimal

places.

**Answer:**

** Laws of Exponents **

•Theorem. [Laws of Exponents]

If s, t, a and bare real numbers with a>0

and b>0, then

** Exponential Functions **

•Exponential function: function of the

form

•where a is a positive real number (a>0)

•a≠1.

•Domain of f: Set of all real numbers.

**Warning!** This is **not** the same as a power

function.

(A function of the form f(x) = x^{n})

•Theorem.

For an exponential function

f(x) = a^{x}, a >0, a≠1, if x is any

real number, then

**Graphing Exponential **

Functions

•Example.

**Problem:** Graph

**Answer:**

** Properties of the **

Exponential Function

•Properties of , a >1

•Domain: All real numbers

•Range: Positive real numbers; (0, ∞)

•Intercepts:

•No x-intercepts

•y- intercept of y = 1

•x-axis is horizontal asymptote as x →-∞

•Increasing and one-to-one.

•Smooth and continuous

•Contains points (0,1), (1, a) and

•Properties of ,
0 <a <1

•Domain: All real numbers

•Range: Positive real numbers; (0, ∞)

•Intercepts:

•No x-intercepts

•y-intercept of y= 1

•x-axis is horizontal asymptote as x →∞

•Decreasing and one-to-one.

•Smooth and continuous

•Contains points (0,1), (1, a) and

**The Number e**

•Number e: the number that the

expression

approaches as n→∞.

•Use e^{x} or exp (x) on your calculator .

•Estimating value of e

•n= 1: 2

•n= 2: 2.25

•n= 5: 2.488 32

•n= 10: 2.593 742 460 1

•n= 100: 2.704 813 829 42

•n= 1000: 2.716 923 932 24

•n= 1,000,000,000: 2.718 281 827 10

•n= 1,000,000,000,000: 2.718 281 828 46

**Exponential Equations**

•If , then u= v

•Another way of saying that the

function
is one-to-one.

•Examples.

(a) **Problem: **Solve

**Answer:**

(b) **Problem:** Solve

**Answer:**

**Key Points**

•Exponents

•Laws of Exponents

•Exponential Functions

•Graphing Exponential Functions

•Properties of the Exponential Function

•The Number e

•Exponential Equations

**Logarithmic **

Functions

Section 4.4

** Logarithmic Functions **

•Logarithmic function to the base a

•a>0 and a≠1

•Denoted by

•Read “logarithm to the base a of x ”or

“base a logarithm of x”

•Defined: if and only if

•Inverse function of

•Domain: All positive numbers (0,∞)

•Examples. Evaluate the fol lowing

logarithms

(a)** Problem:**

**Answer: **

(b) **Problem:**

**Answer:**

(c)** Problem:**

**Answer:**

• Examples. Change each exponential

expression to an equivalent expression

involving a logarithm

(a) **Problem:**

**Answer:**

(b) **Problem:**

**Answer:**

(c)** Problem: **

**Answer:**

• Examples. Change each logarithmic

ex pression to an equivalent expression

involving an exponent.

(a) **Problem:**

**Answer:**

(b) **Problem:**

**Answer:**

(c)** Problem:**

**Answer:**

**Domain and Range of **

Logarithmic Functions

•Logarithmic function is inverse of the

exponential function.

•Domain of the logarithmic function

•Same as range of the exponential

function

•All positive real numbers, (0, ∞)

•Range of the logarithmic function

•Same as domain of the exponential

function

•All real numbers, (-∞, ∞)

•Examples. Find the domain of each function

(a) **Problem:**

**Answer:**

(b) **Problem:**

**Answer:**

**Graphing Logarithmic **

Functions

•Example. Graph the function

**Problem:**

**Answer:**