Find the inverse function of 𝑓(𝑥) = 2𝑒^(2𝑥−1) − 3. State your answer using function notation 𝑓^−1(𝑥)

Question

Find the inverse function of 𝑓(𝑥) = 2𝑒^(2𝑥−1) − 3. State your answer using function notation 𝑓^−1(𝑥)

Answer 1

Hi Avneet,

If we're looking for the inverse function, that means that we're wanting to find a model that has all of the same points as its original function, except their x and y coordinates are switched!

As a result, to find the inverse function, we can do so by simply switching x and y from the original function and solving for y again!

𝑓(𝑥) = 2𝑒^(2𝑥−1) − 3

y = 2𝑒^(2𝑥−1) − 3

First, switch x and y.

x = 2𝑒^(2y−1) − 3

Next, we can first add 3 on both sides to get:

x + 3 = 2𝑒^(2y−1)

Then, we can divide by 2 on both sides and get:

x + 3 = 𝑒^(2y−1)

2

After that, we can take the natural log (ln) of both sides. It would help us here because a natural log is specifically the log with a base e, Euler's number, and that will leave us with 2y - 1 on the right side!

ln((x+3)/2) = 2y - 1

Add one to both sides to get:

ln((x+3)/2) + 1 = 2y

Finally, divide by 2 on both sides to get the inverse function!

f^-1(x) = ln((x+3)/2) + 1

2

If you graph both f(x) and its inverse, you should find that it is symmetric about y = x!

Any questions, let me know.

Thanks!