Video Transcript
Given that 𝑓 of 𝑥 equals four to
the 𝑥, determine the value of 𝑓 of 𝑥 over 𝑓 of 𝑥 minus one minus 𝑓 of 𝑥 minus
one over 𝑓 of 𝑥.
In this problem, we are asked to
find the value of an expression for 𝑥 without being given a value for 𝑥, hinting
that this expression has the same value regardless of the value of 𝑥. We are given that 𝑓 of 𝑥 equals
four to the 𝑥. Therefore, 𝑓 of 𝑥 minus one
equals four to the 𝑥 minus one. Therefore, 𝑓 of 𝑥 over 𝑓 of 𝑥
minus one equals four to the 𝑥 over four to the 𝑥 minus one. Using the quotient rule, this is
equal to four to the 𝑥 minus 𝑥 minus one, which is equal to four to the one which
is equal to four. This is also the value of the base
of the exponential function 𝑓 of 𝑥, which is defined by 𝑓 of 𝑥 over 𝑓 of 𝑥
minus one.
The second part of the expression
is 𝑓 of 𝑥 minus one over 𝑓 of 𝑥, which is just the reciprocal of what we just
found, one over 𝑓 of 𝑥 over 𝑓 of 𝑥 minus one. Therefore, this is equal to one
over four or one-quarter. Therefore, 𝑓 of 𝑥 over 𝑓 of 𝑥
minus one minus 𝑓 of 𝑥 minus one over 𝑓 of 𝑥 equals four minus one-quarter,
which equals 15 over four.
