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Finding the Inverse of a Function

Finding the Inverse of a Function


Finding the Inverse of a Function

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  • Inverse Functions
  • Evaluating an Inverse Graphically
  • Finding the Inverse Algebraically


Inverse Functions

An inverse function undoes the operations performed on variables of a function.  For example, if a number is multiplied by 2, and then 3 is added, we can write this as the function .  The inverse to this function first subtracts 3 from the input value, and then divides by 2, so as to completely undo all operations of the original function.  We write this as: 

If a function becomes the input of an inverse function, then the output is the argument of the original function.  Mathematically, we write this as: 

Evaluating an Inverse Graphically

On a graph, x– and y– coordinates between a function and its inverse inverted, or swapped.  This means that for any coordinate, (x, y), of a function, we can find a corresponding coordinate on the graph of its inverse using the coordinates (y, x).  This means we locate the x–value on the y–axis, and locate the y–value on the x–axis.  This is shown in the graph below:

Finding the Inverse Algebraically

If we want to find the inverse of a function algebraically, there are two common procedures most people use:

  • One method is to rewrite the equation, except swap x and y.  Then, rewrite the equation so that y is isolated on one side of the equation.  
  • The other method is to do the above process in reverse order.  First, you can rewrite the equation so that x is isolated on one side of the equation.  Then, simply swap x and y.  The resulting equation will be the defined inverse function. 

Let's define the inverse of  using one of the methods above: 

Technically, we need to restrict the domain of the inverse function to non-negative values of x.  This is because the range of the original function was restricted to non-negative y–values.  The domain and range of a function also swap when defining the domain and range of its inverse.