Inverse Functions
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 x2 is . Let’s see how “undoes” x2:
For the Math IC, it is 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 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–23.
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.
  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
Contrary to their name, inverse functions are not necessarily functions at all. Take a look at this question:
Is the inverse of f(x) = x3 a function?
Begin by writing y = x3. Next, switch the places of x and y: x = y3. 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:
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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