Graph, using graphing calculators, the two functions 𝑓 sub one of 𝑥 equals one-half to the power 𝑥 minus six and 𝑓 sub two of 𝑥 equals one to find the solution set of the equation one-half to the power 𝑥 minus six equals one.
The first function that we’re given in this question is an exponential function. This simply means that it’s of the form 𝑓 of 𝑥 equals 𝑎 times 𝑏 to the power 𝑥, where 𝑎 and 𝑏 are real constants; 𝑏 is positive and 𝑏 is not equal to zero. We are asked to graph these two functions so that we can find the solution of the equation one-half to the power 𝑥 minus six equals one. We could, of course, graph these two functions on paper, but here we’re told to use graphing calculators.
Don’t worry if you haven’t got a graphing calculator; we can often find graphing software online on websites. When we are inputting these functions onto the graphing calculator or graphing software, it’s very likely that we’ll have to use the 𝑦-variable instead of 𝑓 of 𝑥. Once we have inputted both functions, we can observe the results. Both of these functions plotted on the same graph will look something a little like this. We notice that the exponential function 𝑓 sub one of 𝑥 demonstrates an exponential decay. The other function 𝑓 sub two of 𝑥 equals one is a horizontal line passing through one on the 𝑦-axis.
Let’s now see how we can use the graphs of these functions to solve the given equation. We can remember that the solution set of an equation is the set of all values that satisfy that equation. We’ll need to recognize how this equation is linked to the two functions that are graphed. So, in order to satisfy the equation one-half to the power 𝑥 minus six equals one, we’re looking to see if there are any points or even a single point which lies on both functions. That means we should check if there are any points of intersection.
Looking at the graph, we can see that there is one point of intersection when 𝑥 is equal to six. We know that because we have an exponential function 𝑓 sub one of 𝑥, then the curve is going to continue in these directions. So, it will not meet the function 𝑓 sub two of 𝑥 equals one at any other points. Therefore, we can give the answer that the solution to the equation one-half to the power 𝑥 minus six equals one is the set containing six.