Video Transcript
Use the graph of 𝑓 of 𝑥 equals two to the power 𝑥 minus one to answer the following question. True or False: The equation two to the power of 𝑥 minus one equals two has no solution.
Here we are given the graph, which is that of an exponential function 𝑓 of 𝑥 equals two to the power of 𝑥 minus one. We need to use this graph to help us establish if two to the power of 𝑥 minus one equals two has a solution or not. We can do this by taking the function and establishing if there is an output value 𝑓 of 𝑥, which is equal to two. We can add to the graph the line which represents 𝑓 of 𝑥 equals two. It is a horizontal line passing through two on the 𝑦-axis.
The solution to the equation two to the power of 𝑥 minus one equals two is any points that lie on both functions 𝑓 of 𝑥 equals two to the power of 𝑥 minus one and 𝑓 of 𝑥 equals two. We can see from the graph that there is one point of intersection. It’s at the coordinate two, two. That means that there is a solution to this given equation. Therefore, we can say that this statement is false.
We could even work out using the graph that the solution is the 𝑥-value of the point of intersection. It would be two. We could even solve this equation algebraically if we wished. If we substituted 𝑥 is equal to two into the left-hand side of this equation, we would have two to the power of two minus one. This simplifies to two to the power one, which is equal to two. There are a lot of twos in this question, but we’ve really worked out that when there is an 𝑥-value of two, then this equation is solved.
When we are using graphs to find the solution set of an equation, then there will be no solution set if the two functions do not intersect. And so we have verified that the statement that the equation two to the power 𝑥 minus one equals two has no solution is false.
