Video Transcript
Use the graphs below to answer the
following question. True or false: the equation two to
the power of 𝑥 is equal to negative 𝑥 has no solution.
In this question, we’re given the
graph of two functions. Let’s start by determining which
two functions these are the graphs of. First, we can see that our straight
line passes through the origin, so its 𝑦-intercept is zero. Next, we can see for every one unit
we go across, we travel one unit down. So, its slope is negative one. In slope–intercept form, that’s the
line 𝑦 is equal to negative one 𝑥 plus zero, which is just 𝑦 is equal to negative
𝑥. Our other curve has the shape of an
exponential function, and we can see it passes through the point with coordinates
one, two. If we substitute 𝑥 is equal to one
into the function two to the power of 𝑥, we can see this outputs a value of
two. We could do this with other points
on our curve to conclude that this is indeed a sketch of the curve 𝑦 is equal to
two to the power of 𝑥.
We need to use these graphs to
determine whether or not the equation two to the power of 𝑥 is equal to negative 𝑥
has a solution. We might be tempted to try and
solve this by using manipulation. However, this will be very
difficult because 𝑥 appears in the exponent and not in the exponent. Instead, recall that a solution to
this equation is a value of 𝑥 such that both sides of the equation are equal. In other words, we need to input a
value of 𝑥 into the function two to the power of 𝑥 and then put the same value
into the function negative 𝑥 to get the same output. We can do this directly from our
graph. For the outputs of these two
functions to be equal with the same 𝑥-input, they must have a point of
intersection. This is because the 𝑦-coordinate
tells us the outputs of this function for the given input.
Hence, because there’s one point of
intersection between the line and the curve, we can conclude that two to the power
of 𝑥 is equal to negative 𝑥 has one solution. In fact, we can even approximate
this value by trying to read off its 𝑥-coordinate from the graph. Doing this, we would get that 𝑥 is
approximately equal to negative 0.6. Therefore, we were able to show
that it is false that the equation two to the power of 𝑥 is equal to negative 𝑥
has no solutions.
