Question Video: Solving Exponential Equations Graphically Mathematics • Second Year of Secondary School

Video Transcript

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

In this question, we’re given two functions 𝑓 sub one of 𝑥 and 𝑓 sub two of 𝑥, and we’re given a graph of the function 𝑦 is equal to 𝑓 sub one of 𝑥. We’re asked to find the solution set of an equation. And since 𝑓 sub one of 𝑥 is equal to the left-hand side of this equation and 𝑓 sub two of 𝑥 is equal to the right-hand side of this equation, the equation is 𝑓 sub one of 𝑥 equals 𝑓 sub two of 𝑥. 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 𝑥 and the line 𝑦 is equal to 𝑓 sub two of to 𝑥. Because the point of intersection would have the same 𝑦-coordinate and the 𝑦- coordinate is the output of the function for the given 𝑥 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 𝑥 plus three. First, we note that its 𝑦-intercept will be at three. We can also find its 𝑥-intercept by substituting 𝑦 is equal to zero. Solving this, we get that 𝑥 is equal to negative three. We can then plot our line. Its 𝑦-intercept is at three, and its 𝑥-intercept is at negative three. This then allows us to plot our line. We just connect the 𝑦- and 𝑥-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 𝑥-coordinate; its 𝑥-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 𝑥 is equal to 𝑥 plus three is just the set containing negative one.