Approximating the Solution to an Exponential Equation

Solve \(x+e^{x}=2.\) Express the solution rounded to three decimal places.

Solution We can approximate the solution to the equation by graphing the two functions \(Y_{1}=x+e^{x}\) and \(Y_{2}=2.\) Then we use graphing technology to approximate the intersection of the graphs. Since the function \(Y_{1}\) is increasing (do you know why?) and the function \(Y_{2}\) is constant, there will be only one point of intersection. Figure 69 shows the graphs of the two functions and their intersection. They intersect when \( x\approx 0.4428544\), so the solution of the equation is \(0.443\) rounded to three decimal places.