Question Video: Solving Exponential Equations Graphically Mathematics

The following graph shows the function 𝑓₁(π‘₯) = 2^(βˆ’π‘₯). Use this graph and plot the function 𝑓₂(π‘₯) = π‘₯ + 3 to find the solution set of the equation 2^(βˆ’π‘₯) = π‘₯ + 3.

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Video Transcript

The following graph shows the function 𝑓 sub one of π‘₯ is equal to two to the power of negative π‘₯. Use this graph and plot the function 𝑓 sub two of π‘₯ is equal to π‘₯ plus three to find the solution set of the equation two to the power of negative π‘₯ is equal to π‘₯ plus three.

In this question, we’re given two functions 𝑓 sub one of π‘₯ and 𝑓 sub two of π‘₯, and we’re given a graph of the function 𝑦 is equal to 𝑓 sub one of π‘₯. We’re asked to find the solution set of an equation. And since 𝑓 sub one of π‘₯ is equal to the left-hand side of this equation and 𝑓 sub two of π‘₯ is equal to the right-hand side of this equation, the equation is 𝑓 sub one of π‘₯ equals 𝑓 sub two of π‘₯. We can solve this equation graphically. Any solution to this equation will be a point of intersection between the curve 𝑦 is equal to 𝑓 sub one of π‘₯ and the line 𝑦 is equal to 𝑓 sub two of to π‘₯. Because the point of intersection would have the same 𝑦-coordinate and the 𝑦- coordinate is the output of the function for the given π‘₯ coordinator, which means the outputs of the function would be the same, so our equation would be solved.

We need to sketch the curve 𝑦 is equal to π‘₯ plus three. First, we note that its 𝑦-intercept will be at three. We can also find its π‘₯-intercept by substituting 𝑦 is equal to zero. Solving this, we get that π‘₯ is equal to negative three. We can then plot our line. Its 𝑦-intercept is at three, and its π‘₯-intercept is at negative three. This then allows us to plot our line. We just connect the 𝑦- and π‘₯-intercept with a straight line. Then, the only point of intersection between our line and our curve will be the only solution to our equation. We can read off its π‘₯-coordinate; its π‘₯-coordinate is negative one.

Then, since the question ask us to write this as a solution set, we’ll write this as the set containing negative one. Therefore, we were able to show the solution set of the equation two to the power of negative π‘₯ is equal to π‘₯ plus three is just the set containing negative one.

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