


Inverse Functions
The inverse of a function “undoes” that function. An example
may be the best way to explain what this means: the inverse of x^{2} is . Let’s see how “undoes” x^{2}:
For the Math IC, it is important to know how to find the
inverse of a simple function mathematically. For example:

The easiest way to find the inverse of a function is to
break the function apart step by step. The function f(x)
= 3x + 2 requires that for any value of x,
it must be first multiplied by three and then added to 2. The inverse
of this function must begin by subtracting 2 and then dividing by
three, undoing the original function: f^{–1}(x)
= ^{x}^{–2}⁄_{3}.
You should know how an inverse works in order
to deal with any conceptual inverse questions the Math IC might
throw at you. But if you are ever asked to come up with the inverse
of a particular function, there is an easy method that will always
work.
 Replace the variable f(x) with y.
 Switch the places of x and y.
 Solve for y.
 Replace y with f^{–1}(x).
Here’s an example of the method in action:

First, replace f(x)
with y. Then switch the places of x and y,
and solve for y.
Finding Whether the Inverse of a Function Is a Function
Contrary to their name, inverse functions are not necessarily
functions at all. Take a look at this question:

Begin by writing y = x^{3}.
Next, switch the places of x and y: x = y^{3}.
Solve for y: y = 3. Now you need to analyze the inverse
of the function and decide whether for every x,
there is only one y. If only one y is
associated with each x, you’ve got a function.
Otherwise, you don’t. In this case, every x value
that falls within the domain turns out one value for y, so f^{–1}(x)
is a function.
Here’s another question:

Again, replace x with y and
solve for y:
Now, since you’re dealing with an absolute value, split
the equations:
Therefore,
The inverse of f(x)
is this set of two equations. As you can see, for every value of x except
0, the inverse of the function assigns two values of y.
Consequently, f^{–1}(x)
is not a function.
