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Inverses and Radicals
Inverses and Radicals
Quadratic Function
Arithmetic Sequence and Series
Relation and Function




The inverse of a relation R is the relation consisting of all ordered pairs (y,x) such that (x,y) belongs to R


The inverse of the relation

        (2,3), (4,5), (2,6), (4,6) 


        (3,2),  (5,4), (6,2), (6,4)

Generally we switch the roles of x and y to find the inverse.

For functions, we follow the steps below to find the inverse:

  • Step 1:  Switch the x and y.

  • Step 2:   Solve for y.


  • Step 3:  Write in inverse notation.



Find the inverse of 

        y  =  2x + 1


  1. We write
             x = 2y + 1

  2. We solve:  

            x - 1  =  2y 

                      x - 1
            y  =                  

  3.  We write 

                               x - 1
            f -1(x)   =                   

Notice that the original function took x, multiplied by 2 and added 1, while the inverse function took x, subtracted 1 and divided by 2.  The inverse function does the reverse of the original function in reverse order.


Find the inverse of

  1.               x - 1
    f(x)  =                (x + 1) / (x - 1)
                  x + 1

  2. f(x)  =  3x3 - 2   cubeRt(x/3 + 2/3)

  3. f(x)  =  3x - 4    (x +4)/3

Graphing Inverses

To graph an inverse we imaging folding the paper across the y = x line and copy where the ink smeared in the other side.  


One to One Functions

A function  y = f(x) is called one to one if for every y value there is only one x value with y = f(x).  That is, each y value comes from a unique x value.


  1. Determine if 

            y = 2x - 3

    is one to one


    For any two values a and b if 

            y = 2a - 3 


            y = 2b - 3


            2a - 3  =  2b - 3

            2a  =  2b

            a  =  b

    Hence this function is 1-1.

  2. Determine if 

    y  =  x2

    is one to one


    Now if  

            a2   =   b2  


            a   =   b

    In particular 

            a = -b 

    is a viable solution, for example 

            12  = (-1)2

    hence this function is not 1-1. 

The Horizontal Line Test

If the graph of a function is such that every horizontal line passes through the graph at at most one point then the function is 1-1.  

The graph below is the graph of a 1 -1 function since every horizontal line crosses the graph at most once.


However, the graph below is not the graph of a 1 -1 function, since there is a horizontal line that crosses the graph more than once.




If a function is 1-1, then its inverse is also a function

An application of this is if we want a computer to find an inverse function, then we first have the computer check to see if the function is one to one, then have it proceed to find the inverse.


Simplifying Radicals

When working with the simplification of radicals you must remember some basic information about perfect square numbers.

You need to remember: 

Perfect Squares
4 = 2 x 2
9 = 3 x 3
16 = 4 x 4
25 = 5 x 5
36 = 6 x 6
49 = 7 x 7
64 = 8 x 8 
81 = 9 x 9 
100 = 10 x 10
Radicals (square roots)
= 2
= 3
= 4
= 5
= 6
= 7
= 8
= 9
= 10

While there are certainly many more perfect squares, the ones appearing in the charts above are the ones most commonly used.


To simplify means to find another expression with the same value.  It does not mean to find a decimal approximation.


To simplify (or reduce) a radical:

1.  find the largest perfect square which will divide evenly into the number under your radical sign.  This means that when you divide, you get no remainders, no decimals, no fractions.

Reduce: the largest perfect square that divides evenly into 48 is 16.


If the number under your radical cannot be divided evenly by any of the perfect squares, your radical is already in simplest form and cannot be reduced further.

2.  write the number appearing under your radical as the product (multiplication) of the perfect square and your answer from dividing.

3.  give each number in the product its own radical sign.

4.  reduce the "perfect" radical which you have now created.

5.  you now have your answer.


What happens if I do not choose the largest perfect square to start the process?

If instead of choosing 16 as the largest perfect square to start this process, you choose 4, look what happens.....

 Unfortunately, this answer is not in simplest form.
  The 12 can also be divided by a perfect square (4).

If you do not choose the largest perfect square to start the process, you will have to repeat the process.  



Don't let the number in front of the radical distract you.  It is simply "along for the ride" and will be multiplied times our final answer.

The largest perfect square dividing evenly into 50 is 25.

Reduce the "perfect" radical and multiply times the 3 (who is "along for the ride")

Note:  The examples shown in these lessons on radicals show ALL of the steps in the process.  It may NOT be necessary for you to list EVERY step.  As long as you understand the process and can arrive at the correct answer, you are ALL SET!!




Addition and Subtraction of Radicals 

When adding or subtracting radicals, you must use the same concept as that of adding or subtracting like variables. 
In other words, the radicals must be the same before you add (or subtract) them.

Since the radicals are the same, simply add the numbers in front of the radicals (do NOT add the numbers under the radicals). 
Since the radicals are not the same, and both are in their simplest form, there is no way to combine them.  The answer is the same as the problem.

Warning:  If the radicals in your problem are different, be sure to check to see if the radicals can be simplified.  Often times, when the radicals are simplified, they become the same radical and can then be added or subtracted.  Always simplify, if possible, before deciding upon your answer.


Let's try one together.

1.  Are the radicals the same? Answer:   NO
2.  Can we simplify either radical? Answer:  Yes,  can be simplified.
3.  Simplify the radical. Answer:
4.  Now the radicals are the same and we can add. Answer:



Let's try another one:

1. Simplify  Answer:
2. Simplify Answer:
3. Since the radicals in steps 1 and 2 are now the same, 
we can combine them.
4. You are left with:
5. Can you combine these radicals? Answer:  NO
6. Therefore, Answer: ------------------>


Multiplication and Division of Radicals 

When multiplying radicals, one must multiply the numbers OUTSIDE (O) the radicals AND then multiply the numbers INSIDE (I) the radicals.

When dividing radicals, one must divide the numbers OUTSIDE (O) the radicals AND then divide the numbers INSIDE (I) the radicals.

Let's try an example.

1.  Multiply the outside numbers first 2 • 3 = 6 
2.  Multiply the inside numbers
3.  Put steps 1 and 2 together and simplify
4.  Therefore, the answer is 72.



Let's try another one:

1.  Divide the outside numbers first.
2.  Divide the inside numbers.
3.  Put steps 1 and 2 together and simplify.
4.  Therefore, the answer is:




Math iS cOoL...