Composite Functions Worksheet
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1
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3Composite Functions
What Are Composite Functions?
Composition of functions is when one function is inside of another function. For example, if we look at the
2
function h(x) = (2x – 1)
. We can say that this function, h(x), was formed by the composition of two other
2
functions, the inside function and the outside function. In the case of h(x) = (2x – 1)
, the inside function is
2
2x – 1 and the outside function is z
, the letter z was used just to represent a different variable, we could
have used any letter that we wanted. Notice that if we put the inside function, 2x – 1, into the outside
2
2
2
function, z
, we would get z
= (2x – 1)
, which is our original function h(x).
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The notation used for the composition of functions looks like this, (f g)(x).
So what does this mean
o
(f g)(x),
the composition of the function f with g is defined as follows:
o
(f g)(x)
= f(g(x)), notice that in the case the function g is inside of the function f.
In composite functions it is very important that we pay close attention to the order in which the composition
o
o
of the functions is written. In many cases (f g)(x)
is not the same as (g f )(x).
Let’s look at why the order
is so important:
o
(f g)(x)
= f(g(x)), the g function is inside of the f function
o
(g f )(x)
= g(f(x)), the f function is inside of the g function
o
o
o
(f g)(x)
and (g f )(x)
are often different because in the composite (f g)(x),
f(x) is the outside function and
o
g(x) is the inside function. Whereas in the composite (g f )(x),
g(x) is the outside function and f(x) is the
inside function. This difference in order will often be the reason why we will get different answers for
o
o
(f g)(x)
and (g f )(x).
This means we need to make sure that we pay close attention to the way the
problem is written when we are trying to find the composition of two functions.
How Do You Find the Composition of Two Functions?
Here are the steps we can use to find the composition of two functions:
Step 1: Rewrite the composition in a different form. For example, the composition
o
(f g)(x)
needs to rewritten as f(g(x)).
Step 2: Replace each occurrence of x found in the outside function with the inside function. For
o
example, in the composition of (f g)(x)
= f(g(x)), we need to replace each x found in
f(x), the outside function, with g(x), the inside function.
Step 3: Simplify the answer.
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