## Composite Functions

The function operation that most algebra students dislike is the composite function.  However once algebra students learn to handle these problems they warm up to the topic.  A composite function is nothing more than evaluating a function with another function.  Lets take take the function f(x)=3x + 1.  If I said evaluate the function for x=3 you would write f(3) = 3(3) + 1.  Next you would figure out that 3(3) + 1 is equal to 10 and that would be the answer, f(3) = 10.  Now lets consider the same function but this time lets find f(y +4). Think carefully this would mean that in the function f(x)= 3x + 1 we would let x= (y + 4).  I know this is a little confusing so focus and stay with me.  Now going back to the function just replace the “x” with “(y + 4)” and simplify.  We get f (y + 4) =3(y + 4) + 1 and after we simplify we can write the result as 3y + 3(4) + 1 or 3y + 13.  The idea of evaluating a function with something other than a number is what trips up most students at first- yes it’s a little confusing.  But understanding how do work with functions and variables is key with composite functions.  Lets take that same function above f(x) = 3x + 1  and make up another function g(y) = y + 4.  If I say find f(g(y)) what I’m saying is plug in the g(y) function into the f(x) function and simplify- the process to solve this problem is the same as the example I did above f(y + 4) = 3(y + 4) + 1.  Ok take a deep breath and brush off any sweat, you can do this if you focus and practice- the notes below will help.

some key topics that involve composite functions:

* function operations
* range and domain
* vertical line test
* function inverse