Your Algebra Homework Can Now Be Easier Than Ever!

Exponential and Logarithmic Functions

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 -1are 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: Interchange 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) = xn)

•Theorem.
For an exponential function
f(x) = ax, 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 ex 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 following
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
expression 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:

Prev Next

Start solving your Algebra Problems in next 5 minutes!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath

Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of November 23rd you will pay only $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minute live session from tutor.com for a 1$!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebra Helper is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebra Helper solve it:

Step 3 : Ask for an explanation for the steps you don't understand:



Algebra Helper can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

ORDER NOW!

Algebra Helper
Download (and optional CD)

Only $39.99

Click to Buy Now:


OR

2Checkout.com is an authorized reseller
of goods provided by Sofmath
Check out our demo!
 
"It really helped me with my homework.  I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
 
 
Sofmath
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805
 

Home   : :   Features   : :   Demo   : :   FAQ   : :   Order

Copyright © 2004-2024, Algebra-Answer.Com.  All rights reserved.