Video Transcript
If 𝑓 of 𝑥 is equal to 𝑥 plus two
and 𝑓 of 𝑔 of four is equal to negative three, which of the following could define
𝑔 of 𝑥? Is it A) 𝑔 of 𝑥 equals 𝑥 minus
nine, B) 𝑔 of 𝑥 equals 𝑥 minus two, C) 𝑔 of 𝑥 equals 𝑥 minus three, or D) 𝑔
of 𝑥 is equal to 𝑥 minus four.
Our first step here is to
understand what we mean by the composite function 𝑓 of 𝑔 of 𝑥. This involves substituting our
initial value 𝑥 into the function 𝑔 of 𝑥. We then substitute the answer into
our function 𝑓 of 𝑥. This can be demonstrated using the
diagram shown. We substitute our value of 𝑥 into
𝑔 of 𝑥 and then the answer into the function 𝑓 of 𝑥.
In our question, our value for 𝑥
was equal to four. We were also told that the value of
𝑓 of 𝑔 of four was equal to negative three. Therefore, our final answer is
negative three. We were told in the question that
𝑓 of 𝑥 was equal to 𝑥 plus two. This means that our inverse
function 𝑓 minus one of 𝑥 is equal to 𝑥 minus two as the opposite or inverse of
adding two is subtracting two.
We can work out the number in the
middle box by performing the function 𝑓 minus one of 𝑥. As our final answer was negative
three, we need to calculate 𝑓 minus one of negative three. This is equal to negative three
minus two, which in turn gives us an answer of negative five. The number in the middle box is
negative five.
We could check this answer by
performing the function 𝑓 of negative five. The function 𝑓 of 𝑥 was equal to
𝑥 plus two. Therefore, 𝑓 of negative five is
equal to negative five plus two. This is equal to negative
three. Therefore, the last step of our
calculation is correct.
The function 𝑔 of 𝑥, therefore,
takes us from four to negative five. In order to get from four to
negative five, we need to subtract nine. Four minus nine is equal to
negative five. This means that the function 𝑔 of
𝑥 is equal to 𝑥 minus nine. If 𝑓 of 𝑥 is equal to 𝑥 plus two
and 𝑓 of 𝑔 of four is equal to negative three, then the function that defines 𝑔
of 𝑥 is 𝑥 minus nine. This was option A from the four
options we were initially given.
We can also conclude that the
overall function 𝑓 of 𝑔 of 𝑥 is equal to 𝑥 minus seven. 𝑔 of 𝑥 subtracted nine from our
𝑥-value and 𝑓 of 𝑥 added two to this value. Negative nine plus two is equal to
negative seven.
Our final answer for 𝑓 of 𝑔 of 𝑥
will be seven fewer than our value of 𝑥.
