Video Transcript
The diagram shows the graph of 𝑓
of 𝑥 is equal to two to the power of two 𝑥. Use this graph to find the solution
set of the equation two to the power of two 𝑥 is equal to four.
In this question, we’re given the
graph of an exponential function, and we’re asked to solve an exponential equation
where this function appears. To do this, we start by recalling
the solution set of an equation is the set of all solutions to that equation. This means we’re looking for all of
the values of 𝑥 which solve the equation two to the power of two 𝑥 is equal to
four. To help us do this, let’s start by
replacing two to the power of two 𝑥 in our equation with 𝑓 of 𝑥. This means the equation we’re asked
to solve can be rewritten as 𝑓 of 𝑥 is equal to four.
In other words, we’re just looking
for the values of 𝑥 such that our function 𝑓 outputs a value of four. And we can recall that the
𝑦-coordinate of a point on our curve tells us the output value of a function for
that value of 𝑥. So, we can find all of the values
where our function outputs four by sketching the line 𝑦 is equal to four onto our
diagram. And we can then see on our diagram
there’s only one point on our curve with 𝑦-coordinate equal to four. This is the point of coordinates
one, four.
And it’s worth reiterating this
tells us that 𝑓 evaluated at one is equal to four. And therefore, one is a solution to
our equation. In fact, all solutions to our
equation will be a point of intersection between the line 𝑦 is equal to four and
the curve 𝑦 is equal to two to the power of two 𝑥. So, because we can see there’s only
one point of intersection, we know there’s only one solution. Therefore, the solution set of the
equation two to the power of two 𝑥 is just the set containing one.
