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Another useful function is the inverse function. This function undoes the action of the corresponding original function. Inverse functions obey the following laws

The following is a procedure for finding the inverse of a function. Suppose you have a function defined as f(x). Compute x=f(y) and then solve for y in terms of x. Then you will have

Example 1. Find the inverse of the following function

Solution 1. We first write

Now we solve this in terms of y.

Now we can read off the inverse

We can check this in the inverse by plugging it into the original function

It is possible that your function is too complicated to find its inverse, but the inverse exists just that you can’t find a general formula for it. A numerical solution may be necessary.

Good luck finding the inverse of that

It is important not to confuse the terms reciprocal and inverse. The reciprocal is one divided by a function $latex [f(x)]^{-1}&s=1$ and the inverse is given when solving by the above method for $latex f^{-1}(x)&s=1$.

Example 2. Find the inverse of $latex f(x) = \frac{1}{1+x}&s=1$ and find the reciprocal also

Solution 2. The reciprocal is easy, just flip the numerator and denominator

The inverse is also easy, just solve $latex x= 1/(1+y)&s=1$ which give $latex yx = 1-x&s=1$ or $latex y = \cfrac{1-x}{x}&s=1$

This means the inverse is $latex f^{-1} (x) = x^{-1} -1 &s=1$.

Example 3. Find the reciprocal and inverse of $latex f(x) = x&s=1$

Solution 3. The reciprocal is when you flip the numerator and denominator.

Notice that the inverse of $latex f(x) = x&s=1$ can be computed as x = y or $latex f^{-1}(x)=x&s=1$. The function x is its own inverse. For that reason, it is sometimes called the identity function. The reciprocal and inverse are different for this case.

Example 4. Find the reciprocal and inverse of $latex f(x) = \frac{1}{x}&s=1$

Solution 4. The reciprocal is just

The inverse of $latex f(x) = 1/x&s=1$ can be found from $latex x = 1/y&s=1$. The inverse of $latex f(x) = 1/x&s=1$ is just $latex f^{-1}(x) = 1/x&s=1$. Don’t get confused and just assume the inverse and reciprocal are just the same for simple expressions like x and 1/x.

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