Inverse Functions
Consider the function f (x) = 5x + 2. What does this function do to the input number x?
First, it multiplies the input number by 5, and then it adds 2 to the resulting product.
Now, suppose you were told that 27 was the resulting output number. What would you do to find out what the input number was? First, you would have to subtract 2 then divide by 5 (getting 5). That is, you would have to undo the operations that f did, and in the reverse order. This function that subtracts two then divides by five is called the inverse of f, and is written f1(x). It is the function that undoes what f does.
Now consider the function, the squaring function. How would one undo the squaring function? By taking the square root? What if the input number is - 3? Then g (- 3) = 9. But if we take the square root of 9, we get 3, not - 3. The problem is that there are two input numbers, - 3 and + 3, with the same output number 9. Thus, there is no way to figure out which input number, - 3 or + 3, produced the 9, so there is no way to undo what g did. Remember, a function must always give the same output for any given input. It cannot sometimes give an output of 3 when the input is 9 and other times give an output of - 3 when the input is 9. Thus there is no inverse function for g. This dilemma will always occur for functions that transform two or more input numbers into the same output number.
Read the explanation below to learn more about inverse functions.
Practice: Find the inverse of a function in the following exercises.