1.11. Function Composition http://www.ck12.org
There are two notations used to describe function composition. In each case the order of the functions matters
because arithmetically the outcomes will be different. Squaring a number and then doubling the result will be
different from doubling a number and then squaring the result. In the diagram above,f(x)occurs first andg(x)occurs
second. This can be written as:
g(f(x))or(g◦f)(x)
You should read this “goffofx.” In both cases notice that thefis closer to thexand operates on thexvalues first.
In the following three examples you will practice function composition with these functions:
f(x) =x^2 − 1
h(x) =xx−+^15
g(x) = 3 ex−x
j(x) =√x+ 1
Example A
Showf(h(x)) 6 =h(f(x))
Solution:
f(h(x)) =f(xx−+^15 )=(xx−+^15 )^2 − 1
h(f(x)) =h(x^2 − 1 ) =((xx^22 −−^11 )+)−^15 =xx^22 −+^24
In order to truly show they are not equal it is best to find a specific counter example of a number where they
differ. Sometimes algebraic expressions may look different, but are actually the same. You should notice that
f(h(x))is undefined whenx=−5 because then there would be zero in the denominator.h(f(x))on the other hand
is defined atx=−5. Since the two function compositions differ, you can conclude:
f(h(x)) 6 =h(f(x))
Example B
What isg(h(x))?
g(h(x)) =g(x− 1
x+ 5)
= 3 e(xx−+^15 )−(x− 1
x+ 5)
=3 exp(x− 1
x+ 5)
−
(x− 1
x+ 5)
Note that it is difficult to write and read an exponential function with a large fraction in the exponent. In order to
make it easier to work with you can use ex p(x)instead of exwhich allows more space and easier readability.
Example C