Rational+Functions

This is a discussion on Rational Functions. Click [|here] to see video on factoring.

We love Rational Functions!

This is the second extra credit assignment available for students who are not passing Advanced Math. The guidelines are the same as was with the Polynomial section. http://amosam.wikispaces.com/Discussion+Forums This is to be completed by Monday, May 27th. Use examples and an external link. The topics to write on are: 1. Finding the various asymptotes of rational functions and give examples. 2. Identitfying the graphs of rational functions with the numerator being constant and expalin characteristics. 3. Identifying the range and domain of rational, irrational and absolute value functions. 4. Graphing a rational function which has an oblique asymptote. Expain and work out an example showing all steps. 5. Graphing a rational function which is a line with a hole in it and show the steps with explanation in working out an example. 6. Graphing a rational function with vertical and horizontal asymptotes where the numeratorr gives x-intercepts. Show steps with explanation in doing an example. 7. Simplifying rational expresssions for adding,subtracting, multiplying and dividing. 8. Solving inequalities of rational expresssions. 9. Finding the inverse of functions. 10. Graphing transformations of absolute and irrational functions. 11. Solving absolute value inequalities. 12. Solving irrational equations. 13. Finding discontinuity for irrational functions.

Pick your topic as quick as possible and post it on this page. Once a topic is chosen then no one else can select it. Your write up is to be completed by Sunday, May 26th and the reply postings are to be done by Monday, May 27th. No exceptions!

=1. Solving Irrational Equations (Destiny)= What is an irrational equation? An **irrational equation** is an equation that involves square-roots; also known as a **radical equation**. When solving an irrational equation, one is looking to see if it contains any extraneous solutions. An **extraneous solution** (or root) is a simplified version of an equation that does not satisfy the original equation.

To solve an irrational equation, follow these steps: 1. Isolate the square-root. 2. Square both sides. 3. Solve for "x". 4. Check for extraneous roots. __If the equation has several radicals__, repeat the first two steps of the process to remove all of them.



x – 1 = (x – 7)(x – 7) x – 1 = x^2 – 14x + 49 x – 1 = x^2 – 14x + 49 0 = x^2 – 15x + 50 0 = (x – 5)(x – 10) x = 5 x = 10

<span style="font-family: Arial,Helvetica,sans-serif;">x = 5: <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">x=10: <span style="font-family: Arial,Helvetica,sans-serif;">

That means that 10 is extraneous.

Here's an [|example] containing 2 radicals.

Is 10 extraneous?.. Because if you plug your numbers back in.

5-1=(5-7)(5-7) 4=(-2)(-2) 4=4

10-1=(10-7)(10-7) 9=(3)(3) 9=9

So they both work out to equal each other. Therefore there is no extraneous.

If one equation happened not to work out, that would be your extraneous root.

= <range type="comment" id="434601902_4">Identifying the graphs of rational functions with the numerator being constant and explain characteristics</range id="434601902_4"> (Taylor) =

y= 1/x has lines on opposite quadrants and approaches both a y and x asymptote.

y=1/(x^2) lines on parallel quadrants. both approach an x and y asymptote.

Similar to your orginal 1/x equation, except the left side of the graph is flipped.

y=1/((x^2)-1) Three lines. Two in parallel quadrants and a third in a quadratic shape.

y=1/((x^2)-1) a bell shaped curve with a y asymptote.

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=Simplifying rational expressions for adding, subtracting, multiplying and dividing (Richard)= For addition and subtraction Step 1. When adding or subtracting rational functions you have to first find a common denominator as with regular functions. Step 2. Now you have to multiply the top of the two fractions. Step 3. Add the two different numerators together or with subtraction simply subtract the numerators. Step 4. you have to expand the denominator.

For multiplication and division Step 1. When multiplying or dividing rational functions you have to factor the numerator and denominator Step 2. Remove any common factors that you can Step 3. Simplified

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Example of addition and subtraction: __Find the common denominator__

__Now multiply the top of the functions__

__Now add the numerators together__ __And finally multiply the denominators together and it's simplified__

= **10. <range type="comment" id="434601902_3">Graphing transformations of absolute and irrational functions</range id="434601902_3"> (Chris Berry)** =

When graphing any function, first consider what transformations have occurred to the original function. For example:

f(x) = |-x^2 + 4| -Due to the x^2 we know the graph will be a parabola -When we factor our equation we find the x-intercepts to be x=2 and x=-2 -Because of the negative sign in front of the x there will be a reflection in the graph -When setting x to 0 we get a y intercept of 4

Normally, this would be a reflected parabola with a vertical translation of 4. However since this is an absolute value function, something else occurs, and your graph will look like this:

http://www.mathsisfun.com/graph/function-grapher.php?func1=abs(-x^2+4)&xmin=-10&xmax=10&ymin=-6.16666666666667&ymax=7.16666666666667 As you've probably noticed, instead of passing through our x-intercepts at -2 and 2, it is now “bouncing” one could say. Instead of our point being (3, -5) the point will be (3, 5). The definition of an absolute function is the distance a number is from 0. Because of this, nothing will be below the x-axis because y can never be negative.

When dealing with irrational functions we will do the same as absolute functions, and consider what transformations as occurred to √ x when knowing that the original function √ x looks like this:

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If graphing: f(x) = 3± √ x+4 We know: -It has a horizontal translation of -4 -vertical translation of +3

Don't forget that the 3 is also our restriction. This equation is not a function until it is restricted.

Therefore our function will be: f(x) = 3± √x+4, x ≥ 3 When finished graphing it will look like this: []

= = =8. Solving inequalities of rational expressions (Brandan Trainor)=

<span style="font-family: Arial,Helvetica,sans-serif;">A rational inequality is an inequality which contains a rational expression. When solving <span style="font-family: Arial,Helvetica,sans-serif;"> these rational inequalities, there are steps that lead us to the solution.

<span style="font-family: Arial,Helvetica,sans-serif;"> These values will create intervals on the number line. || <span style="font-family: Arial,Helvetica,sans-serif;"> satisfies the inequality. (Find the intervals which satisfy the inequality). || <span style="font-family: Arial,Helvetica,sans-serif;"> the inequality. ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">To solve Rational Inequalities: ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">1. Write the inequality as an equation, and solve the equation. ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">2. Determine any values that make the denominator equal 0. ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">3. On a number line, mark each of the critical values from steps 1 and 2.
 * <span style="font-family: Arial,Helvetica,sans-serif;">4. Select a test point in each interval, and check to see if that test point
 * <span style="font-family: Arial,Helvetica,sans-serif;">5. Mark the number line to reflect the values and intervals that satisfy
 * <span style="font-family: Arial,Helvetica,sans-serif;">6. State your answer using the desired form of notation. ||

<span style="font-family: Arial,Helvetica,sans-serif;">Example 1:


 * <span style="font-family: Arial,Helvetica,sans-serif;"> __x - 4__ < 4 **
 * <span style="font-family: Arial,Helvetica,sans-serif;">x + 5 **

<span style="font-family: Arial,Helvetica,sans-serif;">Create an equation. Change < to =, and solve. Notice that if x = -5, the denominator is 0.


 * <span style="font-family: Arial,Helvetica,sans-serif;">__x - 4__ = 4 **
 * <span style="font-family: Arial,Helvetica,sans-serif;">x + 5 **

<span style="font-family: Arial,Helvetica,sans-serif;">Multiply both sides by (x + 5) to eliminate the fraction.


 * <span style="font-family: Arial,Helvetica,sans-serif;">(__x - 5__) __x - 4__ = 4 (x + 5) **
 * <span style="font-family: Arial,Helvetica,sans-serif;"> 1 x + 5 **


 * <span style="font-family: Arial,Helvetica,sans-serif;">x - 4 = 4x + 20 **
 * <span style="font-family: Arial,Helvetica,sans-serif;"> - 20 - 20 **


 * <span style="font-family: Arial,Helvetica,sans-serif;">x - 24 = 4x **
 * <span style="font-family: Arial,Helvetica,sans-serif;">- x - x **


 * <span style="font-family: Arial,Helvetica,sans-serif;">__- 24__ = __3x__ **
 * 3 3**


 * <span style="font-family: Arial,Helvetica,sans-serif;">x = - 8 **

<span style="font-family: Arial,Helvetica,sans-serif;">Critical values are:
 * <span style="font-family: Arial,Helvetica,sans-serif;">x = - 8 and - 5 **

<span style="font-family: Arial,Helvetica,sans-serif;">On the number line, plot -5 and -8. Since -5 cannot be used, it is an open circle. The inequality is strictly "less than", so the -8 is also an open circle. <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">Test points:


 * <span style="font-family: Arial,Helvetica,sans-serif;">x = - 9, - 6, and 0 **

__**<span style="font-family: Arial,Helvetica,sans-serif;">x = - 9 **__


 * <span style="font-family: Arial,Helvetica,sans-serif;">__- 9 - 4__ < 4 **
 * <span style="font-family: Arial,Helvetica,sans-serif;">- 9 + 5 **


 * <span style="font-family: Arial,Helvetica,sans-serif;">__-13__ < 4 **
 * - 4**

True
 * 3 1/4 < 4**

__**<span style="font-family: Arial,Helvetica,sans-serif;">x = - 6 **__


 * <span style="font-family: Arial,Helvetica,sans-serif;">__- 6 - 4__ < 4 **
 * <span style="font-family: Arial,Helvetica,sans-serif;">- 6 + 5 **


 * <span style="font-family: Arial,Helvetica,sans-serif;">__-10__ < 4 **
 * - 1**

False
 * 10 < 4**

__**<span style="font-family: Arial,Helvetica,sans-serif;">x = 0 **__


 * <span style="font-family: Arial,Helvetica,sans-serif;">__0 - 4__ < 4 **
 * <span style="font-family: Arial,Helvetica,sans-serif;">0 + 5 **


 * <span style="font-family: Arial,Helvetica,sans-serif;">__- 4__ < 4 **
 * 5**

True
 * - 4/5 < 4**



<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">Stated as an inequality, the solution is:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">**x < - 8** or **x > - 5**

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">Stated in interval notation, the solution is:

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">(**- oo, -8**) **U** (**- 5, + oo**)

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">When the numerator of the inequality is a quadratic expression, combine the <span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;"> Quadratic Inequality method of solution with this Rational Inequality method.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">Example 2:


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__x /\ 2 - 2x - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> x - 2 **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">Separate the numerator, form an equation, and solve this quadratic equation.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">**x /\ 2 -2x - 15**

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">Factor, and find the solutions or critical values for the numerator. Also keep in mind that the denominator has **x = 2** as a critical value as well.


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">x /\ 2 -2x - 15 = 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">(x - 5) (x + 3) **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 5 - 3 **

<span style="font-family: Arial,Helvetica,sans-serif;">Critical values are :
 * x = - 3, 2,** **and 5**

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14.4px;">Place the critical values on a number line. Since the inequality is greater than or equal to, the **x = 5** or **x = - 3** are drawn on the number line as a solid circle, which means to include them as part of the answer. Sine **x = 2** creates an undefined expression, it is drawn as an open circle on the number line.



<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">Test points:


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">x = - 4, 0, 3, and 6 **


 * __<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">x = - 4 __**


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__- 4 /\ 2 - 2(- 4) - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> - 4 - 2 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__16 + 8 - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> - 6 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> __9__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- 6 **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">False
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- 3/2 __>__ 0 **


 * __<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">x = 0 __**


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__0 /\ 2 - 2(0) - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 0 - 2 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__- 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> - 2 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> __15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 2 **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">True
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">7 1/2 __>__ 0 **


 * __<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">x = 3 __**


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__3 /\ 2 - 2(3) - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 3 - 2 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__9 - 6 - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 1 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> __- 13__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 1 **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">False
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- 13 __>__ 0 **


 * __<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">x = 6 __**


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">__6 /\ 2 - 2(6) - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 6 - 2 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> __36 - 12 - 15__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> 4 **


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;"> __9__ __>__ 0 **
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">4 **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">True
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">2 1/4 __>__ 0 **



<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The solution is:


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">- 3 __<__ x __<__ 2 or 5 __<__ x __<__ oo **

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">In interval notation, the solution is:


 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">[- 3, 2) U [5, oo) **


 * Extra Help:**


 * @http://www.youtube.com/watch?v=w3GiVaJyvro**


 * @http://www.youtube.com/watch?v=SJecFvUbJOY**

@http://www.youtube.com/watch?v=ZjeMdXV0QMg

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=**__Finding the inverse of functions__ - Ryan**=


 * An invertible function is a function that can be inverted. An invertible function must satisfy the condition that each element in the domain corresponds to one distinct element that no other element in the domain corresponds to. That is, all of the elements in the domain and range are paired-up in monogomous relationships - each element in the domain pairs to only one element in the range and each element in the range pairs to only one element in the domain. Thus, the inverse of a function is a function that looks at this relationship from the other viewpoint. So, for all elements a in the domain of f(x), the inverse of f(x) (notation: f-1(x)) satisfies:**


 * f(a)=b implies f-1(b)=a**


 * And, if you do the slightest bit of manipulation, you find that:**


 * f-1(f(a))=a**


 * Yielding the identity function for all inputs in the domain. When we graph functions and their inverses, we find that they mirror along the line x=y. This is only logical. From our definition, we know that for each (a,b) in f(x) there will be a (b,a) in f-1(x):**


 * Find the inverse (example) :**
 * f(x) = 8 / 9 - 3x**
 * x= 8 / 9-3y**
 * (9 - 3y)x = 8**
 * 9 - 3y = 8 / x**
 * -3y = (8 / x) -9**
 * y = (8 / x) - 9 all over -3**
 * f^-1(x) = 8 / x -9 all over -3**

[|How To Video.] [|Extra help]

=Graphing a rational function which has an oblique asymptote (Spencer)=

Finding the various asymptotes of rational functions and give examples. Lewis

The definition of a asymptote is... <span style="background-color: #ffffff; color: #212121; font-family: arial,sans-serif; font-size: small;">A line that continually approaches a given curve but does not meet it at any finite distance.

Here is how its done.

y= x*2-5x+6 x-1 Factor the top y=(x-3)(x-2) x-1

So because the top is greater then the bottom there is no horizontal asymptote, but there is a vertical and oblique. To find the oblique asymptote you use long division (short cut preferably/synthetic division). You will get 1-4 or x-4 REMEMBER you will have extra numbers after x-1 this is good, this means it is an oblique line. The equation for it is y=x-4 (it will be a diagonal line starting at -4 on the vertical axis and 4 on the horizontal axis. Your oblique asymptote is similar to a linear equation.

Vertical asymptotes are found by setting the denominator equal to zero. x-1=0 x=1

Now for horizontal asymptotes we will use the previous numerator but different denominator say y=x*2-5x+6 x*2+3x+2 Factor both y=(x-3)(x-2) (x+2)(x+1)

So because the top is the same as the bottom their will be vertical and horizontal asymptotes. Vertical is found by setting the denominator equal to zero. x+2=0 x+1=0 x=-2 x=-1

For horizontal asymptotes the same method is used except this time for the numerator. x-3=0 x-2=0 x=3 x=2

That is how you find your 3 asymptotes.

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