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Inverse Functions

Inverse Functions

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Inverse Functions

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Tutorial

  • Inverse Functions and Notation
  • Reverse Operations to Write Inverse Functions
  • The Graph of a Function and its Inverse

Inverse Functions and Notation

In a function, the input variable undergoes one or more operations to yield a specific output value.  We can think of this ideas as x being the input to f(x), where the output value is y.  We can say that f(x) = y. 

With inverse functions, the operations performed on the input variable is undone.  This means that when y is put into the inverse function, the result will be the x value that produced y in the original function.  

We can write this using inverse notation as: f to the power of negative 1 end exponent left parenthesis f left parenthesis x right parenthesis right parenthesis space equals space x

The above expression is telling us that if the value of a function, f(x), goes into an inverse function, f-1(x), the result will be the argument of the original function, x. 


Reverse Operations to Write Inverse Functions

We can write an inverse function by applying inverse operations in reverse order, according to operations that make up a function.  Consider the following example:


Let's work backwards, and apply inverse operations.  This will lead to our inverse function:


 

The Graph of a Function and its Inverse

Below is the graph of f left parenthesis x right parenthesis space equals space 2 x plus 3 and f to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space space fraction numerator x minus 3 over denominator 2 end fraction

 

Notice that there is symmetry between the graph of a function and its inverse.  The line y = x acts as a line of reflection between a function and its inverse.  This is characteristic of all functions and their inverses.


The line y = x is a line of reflection between a function and its inverse.  This means that if you have the graph of a function, you can also graph the line y = x, and use this line to plot points on the inverse function.