Composition of Functions
A composition of functions is when you combine two separate functions so that you use the output from one function for the input of the second function. (Or the range (y-values) of one function becomes the domain (x-values) of the second function.
A composition of functions looks like this:
When reading this out loud, you say "f of g of x"
It is important to remember to go in order across from the inside out when using a composition of functions.
When solving for f(g(x)) you first plug in your g(x) function into your f(x) function. For example, if you were given f(x)=5x and g(x)=10x+3, then your first step would be to plug g(x) into f(x).
This would look like f(g(x))=f(10x+3). Can you see how the function g(x) has taken the place of the x in the function f(x)?
The next step would be to plug the f(x) function into what you already have. So now you take your 5x and put it into the equation.
This would look like f(g(x))=5(10x+3)
=50x+15
Unless you are given an x value to work with, this is as far as you can solve the problem.
You can also reverse the order of the equations and be asked to solve g(f(x))
Given the same functions as above and following the same procedure, you would find that after the first step g(f(x)) would look like g(5x)
In the second step, it would then be turned into g(f(x))=10(5x)+3
=50x+3
It is clear that the answers to the functions will be different depending on the order of the function.
Because of this it is extremely important to make sure you keep the functions in the correct order and plug in using the correct order.
Let's try another example
Now find g(f(x)
g(f(x))=g(4x+7)
=3(4x+7)
=12x+21
=12(3)+21
=36+21
=57
Now you know how to compose two functions! Good luck!
nice, job, katie. clear and succint!
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