The following graph shows the
function 𝑓 sub one of 𝑥 is equal to two to the power of negative 𝑥. Use this graph and plot the
function 𝑓 sub two of 𝑥 is equal to 𝑥 plus three to find the solution set of the
equation two to the power of negative 𝑥 is equal to 𝑥 plus three.
In this question, we’re given two
functions 𝑓 sub one of 𝑥 and 𝑓 sub two of 𝑥, and we’re given a graph of the
function 𝑦 is equal to 𝑓 sub one of 𝑥. We’re asked to find the solution
set of an equation. And since 𝑓 sub one of 𝑥 is equal
to the left-hand side of this equation and 𝑓 sub two of 𝑥 is equal to the
right-hand side of this equation, the equation is 𝑓 sub one of 𝑥 equals 𝑓 sub two
of 𝑥. We can solve this equation
graphically. Any solution to this equation will
be a point of intersection between the curve 𝑦 is equal to 𝑓 sub one of 𝑥 and the
line 𝑦 is equal to 𝑓 sub two of to 𝑥. Because the point of intersection
would have the same 𝑦-coordinate and the 𝑦- coordinate is the output of the
function for the given 𝑥 coordinator, which means the outputs of the function would
be the same, so our equation would be solved.
We need to sketch the curve 𝑦 is
equal to 𝑥 plus three. First, we note that its
𝑦-intercept will be at three. We can also find its 𝑥-intercept
by substituting 𝑦 is equal to zero. Solving this, we get that 𝑥 is
equal to negative three. We can then plot our line. Its 𝑦-intercept is at three, and
its 𝑥-intercept is at negative three. This then allows us to plot our
line. We just connect the 𝑦- and
𝑥-intercept with a straight line. Then, the only point of
intersection between our line and our curve will be the only solution to our
equation. We can read off its 𝑥-coordinate;
its 𝑥-coordinate is negative one.
Then, since the question ask us to
write this as a solution set, we’ll write this as the set containing negative
one. Therefore, we were able to show the
solution set of the equation two to the power of negative 𝑥 is equal to 𝑥 plus
three is just the set containing negative one.