### Video Transcript

If 𝑔 of 𝑥 is the inverse of the function 𝑓 of 𝑥, which is equal to two 𝑒 to the power of 𝑥 plus one, find 𝑔 of 𝑥.

In this question, we are given an exponential function two 𝑒 to the power of 𝑥 plus one. We are asked to find its inverse. We recall that a natural logarithmic function is the inverse of an exponential function with a base of 𝑒. If 𝑓 of 𝑥 is equal to 𝑒 to the power of 𝑥, then the inverse of 𝑓 of 𝑥 is equal to the natural logarithm of 𝑥. We can rewrite our function as 𝑦 is equal to two 𝑒 to the power of 𝑥 plus one. And we recall that to find the inverse of a function, we exchange the variables 𝑥 and 𝑦 and then solve for 𝑦.

We need to rearrange the equation 𝑥 is equal to two 𝑒 to the power of 𝑦 plus one to make 𝑦 the subject. Dividing our equation by two, we have 𝑥 over two is equal to 𝑒 to the power of 𝑦 plus one. Taking the natural logarithm of both sides, we have 𝑦 plus one is equal to the natural logarithm of 𝑥 over two. We can then subtract one from both sides of this equation such that 𝑦 is equal to the natural logarithm of 𝑥 over two minus one.

Replacing 𝑦 with 𝑔 of 𝑥, we have the inverse function 𝑔 of 𝑥, which is equal to the natural logarithm of 𝑥 over two minus one. This is the inverse of the function 𝑓 of 𝑥, which is equal to two 𝑒 to the power of 𝑥 plus one.