6.1 The Exponential Function and Its Inverse

The general form of exponential function is y = b^x

There are 2 forms if exponential function. First, y = b^x where b>0, b≠1


For example, the graph below:

The second form of exponential function is y = b^x where b>0


For example, the graph below:
 

KEY FEATURES

Domain : {x ∈ R}


Range: {y = R, y>0}

y-intercept of 1

horizontal asymptote at y= 0


In conclusion, KEY FEATURES of exponential function are




 - it has a repeating pattern of inite differences
 - in the interval of b>1, the rate of change is increasing proportional to the function
 - whereas, in the interval of 0<b<1, the rate of change is decreasing proportional to the function. 

Now you know all the key features of exponential function. Here we move on to the inverse of exponential function!
The inverse function of y = b^x  x = b^y

If you observe it, you actually can find that inverse exponential is reflected at the line y = x.

Please take notes that it's KEY FEATURES are different from exponential function.

Domain: { x ∈ R, x > 0 }

Range: {y ∈ R}

Vertical Asymptote at x = 0

For more information about logarithm and it's inverse, go to

this link, hope it helps! 

Click me! :)

It's back and for better!

Hey, hey, hey~ Yu Hang's back for Advanced Function's blog!

You know what, I don't really like Logarithm... I used to have fear with it. When i was form 4 and form 5 i had a great trauma with it. I used to have a mindset that it's real tough and i just don't happen to understand what LOGARITHM means to me.

Even when we started Chapter 6. I had a bit of nervous~ But, things don't really go too scary, creepy, haunting.... as i think! haha! I guess, my mind GREW! yeah!! maybe... Ms. Joanne's methodology works on me! anyway, I'm happy.

Logarithm? no wooooories~ come and fight! ACHA! HIYAK! 





This is what I feel now. I failed quite some times last time when I was doing my Logarithm chapter in high school. But, now, I believed that I know it better!

Come on, let's go through the chapter 6 - Exponential and Logarithmic Function together! I hope I can give you some helpful information! Have a great day~

< > Inequalities ≤ ≥

This subtopic reminds me a lot of how my add math teacher, taught me doing the inequalities in the quadratic function. I used to forget and mix up with the sign changing of the inequality sign when negative turns positive and vice versa. Sometimes, I even felt Advanced Functions is a continuous of my secondary school add math class and it is also a subject that brings back a lot of beautiful memories of mine when I was in secondary school. But, one thing I prefer Advanced Functions is that it is way more interesting than add math. last time i used to do piles and piles of homework and practices. But, now things changed. We do mind maps and online activities. It's fun!!! That's why I'm doing my homework now... actually I should say blogging... hahaha! 


OK! Enough for my the rubbish-like introduction take make a little sense but not too much sense... OMG... what I'm saying!  


In this subtopic called "Solve Inequalities Using Technologies" I found out, we don't really use much technology like graphing calculator. But, we use different ways like number line method, graphing method and numeric method and 2 graph method. 


First, the number line method.
For inequality sign "<" the arrow on the number line goes towards left as it is less than certain number.
For inequality sign ">" the arrow on the number line goes towards right as it is greater than certain number. 


Watch this video about Graphing Inequalities on a Number Line.












From the example above we noticed one thing. A fully colored dot and a hollow dot. What is the difference?
a fully colored dot on the number lines mean the sign  "≤" or "≥" depending on the direction of the arrow. A hollow dot means the sign "<" and ">" depending on the direction of the arrow. 


Second, the graphing method or the 2 graph method.


By looking at the graph we can know which part of the graph is above or below the x-axis. This method provides us an image of the range where the y-values are positive or negative and thus giving us the answer of the inequalities. 


If we solve the inequalities using 2 graphs method, we need to make both side of the inequalities to be equal to y then plot 2 graphs. The x-value of the intersection point for both graphs will be the answer for the inequalities.


For a clearer understanding of graphing method, I suggest you to go to this link. http://www.sosmath.com/algebra/inequalities/ineq04/ineq04.html. I think it is pretty informative and clear.


For the last method, Numerical method, I found it the best method. I do not know why, i just like it the most. That's why people says "love is blind"... haha!


First we transform the inequalities into an equation that is equal to 0. Then, we look for its x-intercept. Then we test each interval of the x-intercept. 


Let's say for example the x-intercept are -6 and 2, we can try out when x<-6, the function is either positive or negative. Then the interval between -6<x<2 and x>2. 


Then we look for the interval that is positive or negative, according to the sign of the inequalities in the question, and that interval will be our answer. For example, for question "solve x³+2x-1>0" we will look for the positive part of the function as the sign is greater than and vice versa for question with a less than sign.



Web resources: 


Youtube, Graphing Inequalities on a Number Line, <http://www.youtube.com/watch?v=AjZzACaeIco&feature=player_embedded>


S.O.S Math, Solving Polynomial Inequalities by Graphing, <http://www.sosmath.com/algebra/inequalities/ineq04/ineq04.html>

What?! Polynomial Functions has its Family?

In a family, we have some similarities with our parents and siblings and other relatives. It can be the physical appearance or the characteristic. Same goes to the Polynomial equation of a family! I know, you might think what I am talking about absolute ridiculous rubbish. Hahaha~ chill~


First of all, let me explain what is the families of polynomial functions. A polynomial function that belongs to a family have the same characteristic. Polynomial function of the same family have the same x-intercept, roots, and zeros.


Example:


f(x)=(x+1) (x-1) (x+3) and
f(x)=2(x+1) (x-1) (x+3) 
belong to the same family because they have the same x-intercepts, roots and zeros. 


Example 2: 

Three polynomial equations above are of the same family because they have the same x-intercept of 2 and 5.


To determine the equation of the particular member of the family, the only thing that we require is any one point on the graph. We can substitute the x value or y-value into the function and find out the variable that differs the shape of the graph.


In general, polynomial functions of the same family can be represented in the form, 


y = k( x - a₁ ) ( x - a₂ )...( x - an ), where k ∈ R, k ≠ 0.

Birds of a feather flock together. 
Polynomial equation of a family, share the same x-intercept together. 




See? I told you polynomial functions have their own family. Haha! Enjoy~


Web resource:The Family of Quadratic Functions <http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_08/other_calculator_keystrokes/472_Alg1_9-1B_873823_CFX.pdf>

Zer0, Roots and x-intercept. What's Their Relationship???

After I learned the Chapter 2.3, the question of "what's the difference and relationship between zero, root, and x-intercept?" pop in my mind. Why do they have different names yet they are almost the same????? Weird right? :P


Then. I found out they are actually interrelated and they are same in terms of its value.
Let's say for example the graph below.
 
The x-intercept on the graph is the zero of the graph and the polynomial equation; the zero of the graph is the factor of the polynomial equation; the x-intercept and zero are the roots of the polynomial equation. Thus, the zero, roots and x-intercepts are -1 and 5 (as from the graph above). 


BUT, this is only for those polynomial equation that is factorable. For those polynomial equation that are not factorable, we can find out their zero or roots or x-intercept (if they have roots), we need to use the quadratic formula 

or by using our graphing calculator.


Therefore, as a conclusion:

• the real roots of a polynomial equation P(x)=0 is similar to the x-intercept of the graph P(x) and vice versa.
• A factorable polynomial equation's roots are equal to its zero and thus solving the factor of it.
• BUT, when we come to polynomial equation that is not factorable, we need to solve it using technology or by using graphing method. 

Web resources: Wikipedia, "Zero of a Function" <http://en.wikipedia.org/wiki/Zero_of_a_function>

                        
The Math Page, "The Roots, or Zeros, of a Polynomial"          <http://www.themathpage.com/aprecalc/roots-zeros-polynomial.htm>

                       

The Factor Theorem

When I learn about Factor Theorem, I found four things that I think are the essence of the theorem.

First, the factor theorem itself. A brief and quick summary for it:

x-b is a factor of a polynomial P(x) if and only if P(b)=0.

Similarly, ax-b is a factor of P(x) if and only if P(b/a)=0.

Note: the acronym I learnt in the class is "iff" short form of if and only if. So don't be shock if      you see your Advanced Function teacher write iff! Definitely not spelling error. =D

Example:

x-1 is the factor of P(x)= x³ + 4x² + x - 6 because P(1) = 0

x+1 is not the factor of P(x)= x³ + 4x² + x - 6 because P(-1) = 0

Second, the integral zero theorem. A brief explanation of this theorem:

If x-b is a factor of a polynomial function P(x) with leading coefficient 1 and remaining coefficient that are integers, then b is a factor of the constant term of P(x). 

Thus, as a summary of this theorem, the word zero means the value of b when polynomial function P(x)=0.

Example:

P(x)= x³ + 2x² - 5x - 6

According to the Integral Zero Theorem, the possible factors of the constant term of P(x) MIGHT BE the possible factor in the form of x-b of P(x).

So, the zeroes of the constant term are: ±1, ±2, ±3 and ±6. After that, substitute all those zeroes into P(x). In this case, the zeroes that make P(x)=0 are -3, -1 and 2. Thus, the factors are x+3, x+1, and x-2.

Third, the Rational Zero Theorem. A brief explanation on it:

P(x) is a polynomial function with all integers coefficient and a leading coefficient that is greater than 1 or not an integer, x=b/ is the zero of P(x) where a and b are integers. Then, 

• b is the factor of the constant term,
• a is the factor of the leading coefficient,
• ax-b is the factor of P(x).

Example:

The polynomial function of P(x)= 3x³ + 2x² - 7x + 2 needs Rational Zero Theorem to solve it. 

The factors of the leading coefficient are ±1 and ±3. The factors of the constant term are ±1 and ±2. Thus, the possible combination of zeroes that can make P(b/a) are ±1/1, ±1/2, ±3/1 and ±3/2. In this case, the zeroes that make P(b/a)=0 are 1, -2 and 1/3. Thus, the factors of P(x) are x-1, x+2 and 3x-1.

Fourth, the grouping factorization method. I found this method is quite convenient but it is different from what we (Malaysia national education system's students) learn in form 4 and 5. Don't freak out, you will find it easy after you read the following.


Grouping factorization method can be applied when pairs of term of polynomial can be grouped to factor out a common factor so that the resulting binomial factors are the same.


Example:


f(x)= x³ + 2x² - 9x -18.                 <~ group out the binomial factors
     = x² (x + 2) - 9 (x + 2)             <~ group the remaining numbers
     = (x + 2) ( x² - 9 )                    <~ factorize the factor that has power that is great than 1
     = (x + 2) (x + 3) (x - 3 )           <~ DONE ^^ 

That's it for the Factor Theorem. If you worry I make you blur (definitely I hope I'm not!!!), just youtube-ing the video below. I found it very useful. Thanks for reading and "digesting" what I posted! 

Web resources: Youtube, "The Factor Theorem, ExamSolution", <http://www.youtube.com/watch?v=6G3iAgpK4kA&feature=player_embedded>


Purplemath, "The Factor Theorem", <http://www.purplemath.com/modules/factrthm.htm>

A cute little song to relax~

Hello people, I found a cute little song. I like the last line that says, "Yes try as you may,you just can't get away FROM MATHEMATICS." Personally, I am the type of person who can't get away of maths if I could not get it solved! Is it quite sad to hear. But, no pain, no gain. The more I suffer as a result I can't get it away, the more the satisfaction I got from it once I get it solved! I think all maths freak are the same! gimme 5! piakkkk! XD 


Web resource: Youtube, "Math song", <http://www.youtube.com/watch?v=ZPMRA4yFeeM>