Question Video: Solving Exponential Equations Graphically

Use the given graph of the function 𝑓(π‘₯) = 2^(5 βˆ’ π‘₯) to find the solution set of the equation 2^(5 βˆ’ π‘₯) = 2.

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Video Transcript

Use the given graph of the function 𝑓 of π‘₯ is equal to two to the power of five minus π‘₯ to find the solution set of the equation two to the power of five minus π‘₯ is equal to two.

In this question, we’re given the graph of an exponential function and this exponential function appears in the given exponential equation. We need to use this to determine the solution set of the equation. First, we recall the solution set of an equation is the set of all solutions to that equation. Therefore, we’re looking for the set of all values of π‘₯ which balance both sides of the equation. Another way of thinking about this is since two to the power of five minus π‘₯ is equal to the function 𝑓 of π‘₯, we can substitute 𝑓 of π‘₯ into our equation. This gives us the equation 𝑓 of π‘₯ is equal to two. We’re looking for the set of all values of π‘₯ such that 𝑓 of π‘₯ is equal to two.

To find these values of π‘₯, we can recall that every single point on the curve 𝑦 is equal to 𝑓 of π‘₯ will have coordinates of the form π‘₯, 𝑓 of π‘₯. In other words, the 𝑦-coordinates of the points on the curve tell us the outputs of our function for the given value of π‘₯. We want to determine the values of π‘₯ where our function outputs two. These will be the points on our curve with 𝑦-coordinate equal to two. So, we can find these by sketching the line 𝑦 is equal to two onto the same set of axes. We can see there’s only one point on our curve of 𝑦-coordinate equal to two. It will be the point of intersection between the line 𝑦 is equal to two and the curve 𝑦 is equal to two to the power of five minus π‘₯. The 𝑦-coordinate of this point is two and its π‘₯-coordinate is four. In other words, when π‘₯ is equal to four, our function outputs two. 𝑓 evaluated at four is two.

Therefore, π‘₯ is equal to four is a solution to our equation. In fact, since this is the only point of intersection between the line and the curve, this is the only solution to our equation. This means the solution set to our equation is just the set containing four.

It’s also worth noting we can check our answer by substituting π‘₯ is equal to four into our equation or into our function. Substituting π‘₯ is equal to four into our function 𝑓 of π‘₯, we get 𝑓 evaluated at four is two to the power of five minus four. Five minus four is equal to one. So, this simplifies to give us two to the first power. And any number raised to the first power is just equal to itself. So, 𝑓 evaluated at four is equal to two, which is exactly the same as the right-hand side of our equation, confirming that π‘₯ is equal to four is a solution to our equation. Therefore, we were able to show the solution set of the equation two to the power of five minus π‘₯ is equal to two is just the set containing four.

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