Compositions
By: Ale Mathies
What are compositions?
For two functions, lets say f(t) and g(t) the function
f(g(t)) is said to be the composition of f with g. f(g(t)) is defined by the
output of g(t) and using it as input for f(t).
(Tip) Remember that when you compose a function it is better
to start with the inner function.
For example, in f(g(t)) the inner composition would be g(t)
while f would be the outer function.
There are two ways of solving compositions. Below you will
find, fully explained, both ways.
Way 1:
This way consists of evaluating first and then using
composition
Example:
Let f(x)=2x+1 and
g(x)=x^2 -5
a)
find f(g(3))
(remember to
start with the inner function, so first we work with g(3))
g(3)=(3)^2-5
g(3)=4
then we would
plug it in the outer function
f(g(3))=2(4)+1
f(g(3)=9
Way 2:
Find f(g(x)) first and g(f(x))
first and by finding the composition function first
Let f(x)=x^2-1 and let
g(x)=2x+3
a) Find
f(g(3))
b) f(g(x))=f(x^2-1)
2(x^2-1)+3
2x^2-2+3
2x^2+1
Then find
f(g(3)
F(g(3))=2(3^2)+1
F(g(3))=19
Ale this is a great explanation of compositions it is a great explanation and the examples are very easy to understand great job!!
ReplyDeleteThis makes compositions very easy to follow, great job! I like your examples in this lesson plan, it cleared up any confusions I may have otherwise had. Good post :)
ReplyDeletealejandra,
ReplyDeletekudos to you for tackling compositions! your first example was correct, however, your second example was incorrectly calculated. f(g(x)) = f(2x + 3) = [(2x + 3)^2] - 1, which is 4x^2 + 12x + 9 - 1 = 4x^2 + 12x + 8. so f(g(3)) = 4(3)^2 + 12(3) + 8 = 36 + 36 + 8 = 80.
professor little