Inverse Functions

If f(x) and g(x) are inverses, then

    \[[f \circ g](x)=x\]

    \[[g \circ f](x)=x\]

 

Creating Inverse Functions

We know that when a function is composed into its inverse the result is just x. How do we create inverse functions though?

Step 1: Change f(x) into a y

Step 2: Switch x and y

Step 3: Solve for y. You should get y=.....

Step 4: replace y with f^{-1}(x)

Example: Find the inverse of f(x)=7x-3

  1. y=7x-3
  2. x=7y-3
  3. x+3=7y \longrightarrow y=\frac{x+3}{7}
  4. f^{-1}(x)=\frac{x+3}{7}

 

Graphs of Inverse Functions}

Graphs of inverse functions are always reflections over the line y=x (yes, the same x that’s a solution when we find they’re inverses).

Another way of saying this is that if (a,b) is a point

    \[(a,b) \in f(x) \Leftrightarrow (b,a) \in f^{-1}(x)\]

That’s another way of saying: switch the x and y.

Example 1)

    \[f(x)=5x-4\]


    \[f^{-1}(x)=\frac{x+4}{5}\]

inverse1

              
The values of the points are reversed. There is also a reflection over the line y=x