Given that 𝑓 of 𝑥 equals three 𝑥
plus two, find 𝐵 so that 𝑔 of 𝑥 equals negative three 𝑥 plus 𝐵 satisfies 𝑓 of
𝑔 equals 𝑔 of 𝑓.
Let’s take a moment to understand
the notation being used in this question. We have two functions, 𝑓 of 𝑥 and
𝑔 of 𝑥. We’re then given this equation
here: 𝑓 of 𝑔 equals 𝑔 of 𝑓. On the left-hand side, we have 𝑓
of 𝑔, which means the composite function we get when we take an input, apply the
function 𝑔, and then apply the function 𝑓 to the result, whereas on the right-hand
side we have 𝑔 of 𝑓, which means the composite function we get when we take an
input, apply the function 𝑓 first, and then apply the function 𝑔 to the
The order in which we apply these
functions is really important because in general composition of functions is not
commutative, by which we mean the order does matter. In this specific case of these two
functions 𝑓 and 𝑔, we’re told that 𝑓 of 𝑔 of 𝑥 is equal to 𝑔 of 𝑓 of 𝑥 for a
particular value of this unknown 𝐵. And we need to work out what that
value is. To do this, let’s find algebraic
expressions for 𝑓 of 𝑔 and 𝑔 of 𝑓 of 𝑥.
Let’s start with 𝑓 of 𝑔 of
𝑥. When we take an input 𝑥 and apply
the function 𝑔, we get negative three 𝑥 plus 𝐵. We then take this as the input to
the function 𝑓. So, we have 𝑓 of negative three 𝑥
plus 𝐵. Now, 𝑓 is the function that takes
an input, multiplies it by three, and then adds two. So, we’re going to replace the
input 𝑥 with negative three 𝑥 plus 𝐵. So, we multiply negative three 𝑥
plus 𝐵 by three and then add two. Distributing the parentheses, we
have that 𝑓 of 𝑔 of 𝑥 is equal to negative nine 𝑥 plus three 𝐵 plus two.
Now let’s see what happens when we
compose the two functions in the other order. So, we’re going to take an input
𝑥, apply the function 𝑓, and then apply the function 𝑔. Applying the function 𝑓 first
gives three 𝑥 plus two. And then we take this as our input
to the function 𝑔. We multiply this expression by
negative three and add the unknown 𝐵. Distributing the parentheses, we
have 𝑔 of 𝑓 of 𝑥 is equal to negative nine 𝑥 minus six plus 𝐵.
Now, we want these two expressions
to be equal to each other for a particular value of 𝐵. So, we have negative nine 𝑥 plus
three 𝐵 plus two equals negative nine 𝑥 minus six plus 𝐵. We can see straightaway that the
𝑥-terms cancel, leaving three 𝐵 plus two equals negative six plus 𝐵. Collecting all of the 𝐵s on the
left-hand side of the equation by subtracting 𝐵 from each side gives two 𝐵 plus
two equals negative six. Then, we subtract two from each
side to give two 𝐵 equals negative eight and finally divide both sides of the
equation by two to give 𝐵 equals negative four. We found then that the value of
this unknown 𝐵 such that 𝑓 of 𝑔 of 𝑥 is equal to 𝑔 of 𝑓 of 𝑥 for all values
of 𝑥 is negative four.
This question does also illustrate
a general point, which is that when we compose two linear functions, such as those
we have here, the coefficient of 𝑥 will be the same no matter which order we
compose them in, in this case it’s equal to negative nine, whereas the constant term
will in general be different.
We found that for the given
functions 𝑓 and 𝑔 of 𝑥, the value of 𝐵 that satisfies 𝑓 of 𝑔 of 𝑥 equals 𝑔
of 𝑓 of 𝑥 is negative four.