Inverse Functions
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 x2 is . Let’s see how “undoes” x2:
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:
What is the inverse of f(x) = 3x + 2?
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:
  1. Replace the variable f(x) with y.
  2. Switch the places of x and y.
  3. Solve for y.
  4. Replace y with f–1(x).
Here’s an example of the method in action:
What is the inverse of the function f(x) = ?
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:
Is the inverse of f(x) = x2 a function?
To answer a question like this, you must, of course, first find the inverse. In this case, begin by writing y = x2. Next, switch the places of x and y: x = y2. 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 even-numbered 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:
What is the inverse of f(x) = 2|x – 1|, and is it a function?
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.
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