Inverse Functions

All functions are written in function notation, where f(x) will equal something.

Example: f(x) = 2x +1

If you were to graph this function, f(x) would just be the y coordinate, or the output value, of the function from the input of x.  Just replace f(x) with y:

y = 2x + 1

When you are looking for the inverse of a function, what you are really doing is undoing all of the operations done to x.  When finding the inverse, the same as before will be done by replacing f(x) with y.  To undo the operations to x, simply switch the variables x and y (or if there is more than one x, replace each with a y):

x = 2y + 1

Now, solve for y:

x - 1 = 2y

This answer is correct, however, to write the inverse of a function in correct function notation, substitute y for 

Final Answer:

Here are two examples of graphs of functions and their inverses:

Here is a short video that shows the same steps but to a longer function:

The big point to remember is to switch your x and y values and then solve for x.  Be sure to check if your inverse is actually a function because it is possible that one does not exist!

Hope this helps! -Sofia

1.4- Compositions of Functions

The composition of two functions is when one function is applied to another. This is another way to combine the functions.

Finding the composition of the function  f with the function  g :

 g(x)  is in the domain of f, making it an input.

 f of g of x

Find  f(g(x)) when  f(x) = 2x + 5, and  g(x) = x - 6

The first step: Understanding what the problem is asking is important when solving. Instead of plugging in x for  f(x) , you are now using the function  g(x).  It becomes your new input.

Solving:

 

       f(g(x)) = f(x - 6)

                   = 2(x-6) + 5

                   = 2x - 12 +5

                   = 2x - 7

 

 Evaluating f(g(x)) when x has a specific value such as 1 or 2 is simple now that we already found f(g(x)).

When x = 1,  you simply plug that number in 2x - 7, which gives you an answer of -5.

g of f of x

g(f(x)) consists of the same methods of solving as f(g(x)), because  f and g are still both just functions.

Find g(f(x)) when  f(x) = 2x + 5, and  g(x) = x - 6

       g(f(x)) = g(2x + 5)

                  = 2x + 5 - 6

                  = 2x - 1

 

As you can see, g(f(x)) ≠ f(g(x))  in this case, but sometimes they are equivalent.

Video:  http://www.youtube.com/watch?v=S4AEZElTPDo (This video explains all the concepts I went over about the composition of two functions clearly.)

Identifying Functions 

As well as the composition of functions, there is also identifying functions. The function h is an example used to identify two functions.

For example if h = (2x-7)³, then for f(g(x)),  f = x³ and g = 2x-7

Finding the inner and outer functions is the whole concept of identifying them. f is the outer while g is the inner.

     

           

 


          

Graphs of Functions 1.2

A graph of a function is the collection of coordinate points (x, f(x)), and that:

x is the distance from the y-axis, while f(x) is the distance from the x-axis

The first thing that is important in graphing a function is finding its domain and range

For example:

But in order to find the domain and range of this function, we can not have a negative inside the radical, so the problem must be changed to an inequality in which it would look like this:

0 < -2x+3

Then you would add 2x to both sides, now the expression would look like 2x < 3

To finish the problem and find the domain, divide by 2 on both sides, and the domain would come out to be x < 1.5

The graphical solution would look very similar to this, and because the graph is always going to have a range of equal to, or less than 0.


Increasing and Decreasing Functions

We know if a function is decreasing if,

X₁<X₂ 

implies

and if it's increasing, just switch the inequality sign around.

The function is constant when

Relative Minimum and Maximum Values

A funnction that has a relative minimum of f contains a so that f(a) < f(x)

and that a function with a relative maximum of f contains a so that f(a) > f(x)

Even and Odd Functions

Even Function

Odd Function

A function is even when the x in the domain f is f(-x) = f(x)

A function is odd when the x in the domain f is f(-x) = -f(x)

 

Hopefully this blog was helpful in learning how to graph functions, and finding domain/range.

The Difference Quotient 1.1

The difference quotient easy to apply once you know the steps. For a given expression substitute x+h for your value instead of x and subtract your original expression. 

Ex.  

 

  turns into                                                  

Then distribute the -2 and combine like terms. The end result should look like this.

The answer is -2 but to make this simplified version equivalent to the original you must include 

Ex. 2 

Substitute both 5+h and 5 into the function given. so it looks like this with the red numbers the part substituted in. 

Then Distribute and add like terms until you get to the final answer.

the final answer should look like this.

Thats all there is to doing difference quotients. Hopefully you understand from these two examples but if not there are multiple videos which go over the same topic.

The difference quotient:

http://www.youtube.com/watch?v=1O5NEI8UuHM

Precalculus- Computing Difference Quotients:

http://www.youtube.com/watch?v=AQatV_NJOsA

Pre-Calculus - Evaluating the difference quotient:

http://www.youtube.com/watch?v=OXXrsxA4f_4

Transformations of Functions (1.3)

Transformations of Functions (1.3)


Hey, I will be reviewing the transformations of functions. 

Lets get right to it, its not too difficult but a friendly tip, if the transforming factor added, like a number or fraction etc., to your function is inside of your parenthesis you basically take the opposite of what it should be (multiply by -1).


Translation (Movement)

Outside the parenthesis
For example if we have f(x)+2 the function will move UP 2 units.
If we have f(x)-2 then it will move DOWN 2 units, very logical.

Then we move into the parenthesis...
f(x-2) this will move the function over RIGHT 2 units, not left as you may think.
f(x+2) this follows the trend and will move the function 2 units LEFT.

Stretching

This is where it affects the actual appearance of your graph but hods your intercepts. 

c is any number

f(x) c      This makes your function steeper, but holds all x intercepts in place. Acts like it should, smaller makes the function smaller.

f(x c)       This stretches your function horizontally making a smoother incline and holds all mins and max's in                place as well as the y intercept. acts opposite than you think making less than 1 stretch the function.

Reflecting

Once again 2 types; in and out of parenthesis...

-f(x)    Reflects over the x-axis

f(-x)    Reflects over the y-axis

Those are the transformations of functions, and I hope the graphics help you remember what is what.

Just remember, if it's in the parenthesis, it acts opposite of common sense.

Good luck, Tyler.

Arithmetic Combinations of Functions

Arithmetic Combinations of Functions, Composition of Functions

There are 5 types of arithmetic combinations of functions but we have only been taught 4 as of now. The 4 are sum, difference, product, and quotient.

Sum
(f+g)(x)= f(x) + g(x)

Ex:    

           
             

Difference
(f-g)(x)= f(x) - g(x)

Ex:
   

             
           
              

                  
Product
(f*g)(x)= f(x) * g(x)

Ex:
     
 
            
           
           

Quotient 
 

Ex:
       
                        

             




Now you try!!!

http://www.education.com/study-help/article/pre-calculus-help-combinations-functions/


(f+g)(3) = 0 + (-2) = -2

(f-g)(-5) = -2 - 4 = -6 








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Function Notation Notes

Function Notation Notes

Write in function notation:

y=x^2+x

y=8x-5

f(x)=x^2+x

g(x)=8x-5

Find f(4)

f(4)=(4)^2+4
f(4)=16+4
f(4)=20

Find g(4)

g(4)=8(4)-5
g(4)=32-5
g(4)=27

f(x-3)=(x-3)^2+x-3
f(x-3)=(x-3)(x-3)+x-3
f(x-3)=x^2-3x-3x+9+x-3
f(x-3)=x^2-5x+6

Idea: Plug in a number and replace x with that value

Summary: In function notation, y turns into f(x).  x is the input and f(x) is the output.  You plug x into the equation to evaluate it.

Source: YouTube-- Function Notation (by PatrickJMT)
Picture from:
http://www.google.com/imgres?q=function+notation+graph&um=1&hl=en&client=safari&sa=N&rls=en&biw=1024&bih=614&tbm=isch&tbnid=02fbba1canHOJM:&imgrefurl=http://www.fourcroy.org/algebra2/learning_modules/03_graphs_functions/6_function_notation.html&docid=xeU6XFpZsc4VLM&imgurl=http://www.fourcroy.org/algebra2/pictures/function_basic_graphs.png&w=390&h=281&ei=ScpbUJvGIqHtygGOkoGgAQ&zoom=1&iact=hc&vpx=397&vpy=112&dur=1434&hovh=190&hovw=265&tx=137&ty=183&sig=105214368612422428244&page=1&tbnh=128&tbnw=178&start=0&ndsp=15&ved=1t:429,r:12,s:0,i:111


In a graph, x=the input values and f(x)= the output values