


Inverse Functions
The inverse of a function “undoes” that function. An example
is the best way to help you understand what this means: the inverse
of x^{2} is . Let’s see how “undoes” x^{2}:
You can think of an inverse function as one that takes
the output (y) and returns the input (x).
This way of looking at an inverse helps you see the graphical relationship
between the graph of a function, f(x),
and its inverse, f^{–1}(x).
The graph of the inverse of a function is simply the graph of the
original function reflected across the line y = x.
For the Math IIC, it is also 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 3 and then added to 2. The inverse
of this function must begin by subtracting 2 and then dividing by
3, “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 IIC 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 always works:
 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
Take a look at this question:

To answer a question like this, you must, of course, first
find the inverse. In this case, begin by writing y = x^{2}.
Next, switch the places of x and y: x = y^{2}.
Solve for y: y = . 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. For functions, the square root of a quantity
equals the positive root only; in fact, all evennumbered roots
in functions have only positive values. 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 sample 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.
