Find the Inverse f(x)=3+x+e^x

Problem

ƒ(x)=3+x+ex

Solution

1. Set the function equal to y to represent the output.

y=3+x+ex

2. Interchange the variables x and y to begin finding the inverse function ƒ(x)(−1)

x=3+y+ey

3. Observe that the equation x=3+y+ey involves both a linear term y and a transcendental term ey This type of equation cannot be solved for y using elementary algebraic methods (such as addition, multiplication, or roots).

4. Identify that the inverse function ƒ(x)(−1) exists because ƒ(x) is strictly increasing. Since ƒ(x)′=1+ex which is always positive, the function is one-to-one.

5. Express the inverse using the notation ƒ(x)(−1) While a closed-form expression in terms of elementary functions does not exist, the relationship is defined by the transcendental equation.

Final Answer

ƒ(x)(−1)* is the value *y* such that *x=3+y+ey

Want more problems? Check here!