When learning about inverse functions, students often make some common mistakes that can cause confusion. Here are some things to watch out for:

1. Mixing Up Functions and Their Inverses:

   A common mistake is thinking the original function and its inverse are the same.

   Remember, if you have a function ( f(x) ) that takes an input ( x ) and gives an output ( y ), then the inverse ( f^{-1}(y) ) takes ( y ) and gives you back ( x ).

   For example, if ( f(x) = 2x + 3 ), then the inverse is ( f^{-1}(y) = \frac{y - 3}{2} ).

2. Forgetting About One-to-One Functions:

   A function needs to be one-to-one to have an inverse.

   Sometimes, students forget to check this.

   For example, the function ( f(x) = x^2 ) is not one-to-one because both ( 1 ) and ( -1 ) give the same output of ( 1 ).

   That means its inverse can’t be defined for all real numbers.

3. Making Mistakes When Finding Inverses:

   When trying to find the inverse, students might skip steps or do things the wrong way.

   Here’s how to find the inverse of ( f(x) = 3x - 4 ) correctly:

   • First, replace ( f(x) ) with ( y ): ( y = 3x - 4 ).
   • Next, swap ( x ) and ( y ): ( x = 3y - 4 ).
   • Finally, solve for ( y ): ( y = \frac{x + 4}{3} ).

   This gives us the correct inverse: ( f^{-1}(x) = \frac{x + 4}{3} ).

4. Not Limiting the Domain:

   When working with functions that are not one-to-one, it’s important to limit the domain before finding the inverse.

   For example, if you're looking at ( f(x) = x^2 ), make sure to limit it to ( x \geq 0 ) so that it has a proper inverse.

By keeping these points in mind, students can understand inverse functions better and solve problems more easily!