A. In a composition of two functions the first function is evaluated, and then the second function is evaluated on the result of the first function. In other word, you are going to evaluate the second function in the first function.
Remember that you can evaluate function at any number just replacing the variable in the function with the number. For example, let's evaluate our function at
:
Similarly, to find the composition of and
, we are going to evaluate
at
. In other words, we are going to replace
in
with
:
Remember that two functions are inverse if after simplifying their composition, we end up with just . Let's simplify and see what happens.
Now let's do the same for :
We can conclude that is the inverse function of
, but
is not the inverse function of
.
B. The domain of a function is the set of all the possible values the independent variable can have. In other words, the domain are all the possible x-values of function.
Now, interval notation is a way to represent and interval using an ordered pair of numbers called the end points; we use brackets [ ] to indicate that the end points are included in the interval and parenthesis ( ) to indicate that they are excluded.
Notice that when ,
, so when
,
is not defined; therefore we have to exclude zero from the domain of
.
We can conclude that the domain of the composite function in interval notation is (-∞,0)U(0,∞)
Now let's do the same for .
Notice that the composition is not defined when its denominator equals zero, so we are going to set its denominator equal to zero to find the values we should exclude from its domain:
and
and
Know we know that we need to exclude and
from the domain of
.
We can conclude that the domain of the composition function is (-∞,0)U(0,3)U(3,∞)