find inverse function of y= 3 + x + e^x Solution Let y = 3x + e^x. We want to solve for x. Now, we first divide by 3: y/3 = x + e^x / 3 Then, we take the exponential of both sides. e^(y/3) = e^x * e^(e^x / 3). Then, we divide by 3 again. e^(y/3)/3 = e^x / 3 * e^(e^x / 3) Let z = e^x / 3. Then, we have e^(y/3)/3 = z * e^z which means that z = W(e^(y/3)/3) from the definition of the W function. Or, e^x / 3 = W(e^(y/3)/3). Finally, e^x = 3 W(e^(y/3)/3) and x = ln(3 W(e^(y/3)/3)). Another way to express this result: go back to e^x = 3 W(e^(y/3)/3). Since we know that y = 3x + e^x, we can write, y - 3x = 3 W(e^(y/3)/3) hence, y - 3 W(e^(y/3)/3) = 3x x = (y - 3 W(e^(y/3)/3))/3.

1. find inverse function of y= 3 + x + e^x
Solution
Let y = 3x + e^x. We want to solve for x. Now, we first divide by 3: y/3 = x + e^x /
3 Then, we take the exponential of both sides. e^(y/3) = e^x * e^(e^x / 3). Then, we divide by 3
again. e^(y/3)/3 = e^x / 3 * e^(e^x / 3) Let z = e^x / 3. Then, we have e^(y/3)/3 = z * e^z which
means that z = W(e^(y/3)/3) from the definition of the W function. Or, e^x / 3 = W(e^(y/3)/3).
Finally, e^x = 3 W(e^(y/3)/3) and x = ln(3 W(e^(y/3)/3)). Another way to express this result: go
back to e^x = 3 W(e^(y/3)/3). Since we know that y = 3x + e^x, we can write, y - 3x = 3
W(e^(y/3)/3) hence, y - 3 W(e^(y/3)/3) = 3x x = (y - 3 W(e^(y/3)/3))/3