Video Transcript
Use the given graph of the function
𝑓 of 𝑥 is equal to two to the power of five minus 𝑥 to find the solution set of
the equation two to the power of five minus 𝑥 is equal to two.
In this question, we’re given the
graph of an exponential function and this exponential function appears in the given
exponential equation. We need to use this to determine
the solution set of the equation. First, we recall the solution set
of an equation is the set of all solutions to that equation. Therefore, we’re looking for the
set of all values of 𝑥 which balance both sides of the equation. Another way of thinking about this
is since two to the power of five minus 𝑥 is equal to the function 𝑓 of 𝑥, we can
substitute 𝑓 of 𝑥 into our equation. This gives us the equation 𝑓 of 𝑥
is equal to two. We’re looking for the set of all
values of 𝑥 such that 𝑓 of 𝑥 is equal to two.
To find these values of 𝑥, we can
recall that every single point on the curve 𝑦 is equal to 𝑓 of 𝑥 will have
coordinates of the form 𝑥, 𝑓 of 𝑥. In other words, the 𝑦-coordinates
of the points on the curve tell us the outputs of our function for the given value
of 𝑥. We want to determine the values of
𝑥 where our function outputs two. These will be the points on our
curve with 𝑦-coordinate equal to two. So, we can find these by sketching
the line 𝑦 is equal to two onto the same set of axes. We can see there’s only one point
on our curve of 𝑦-coordinate equal to two. It will be the point of
intersection between the line 𝑦 is equal to two and the curve 𝑦 is equal to two to
the power of five minus 𝑥. The 𝑦-coordinate of this point is
two and its 𝑥-coordinate is four. In other words, when 𝑥 is equal to
four, our function outputs two. 𝑓 evaluated at four is two.
Therefore, 𝑥 is equal to four is a
solution to our equation. In fact, since this is the only
point of intersection between the line and the curve, this is the only solution to
our equation. This means the solution set to our
equation is just the set containing four.
It’s also worth noting we can check
our answer by substituting 𝑥 is equal to four into our equation or into our
function. Substituting 𝑥 is equal to four
into our function 𝑓 of 𝑥, we get 𝑓 evaluated at four is two to the power of five
minus four. Five minus four is equal to
one. So, this simplifies to give us two
to the first power. And any number raised to the first
power is just equal to itself. So, 𝑓 evaluated at four is equal
to two, which is exactly the same as the right-hand side of our equation, confirming
that 𝑥 is equal to four is a solution to our equation. Therefore, we were able to show the
solution set of the equation two to the power of five minus 𝑥 is equal to two is
just the set containing four.
