To understand inverse functions, we need to know what they do before we look at how we write them.

An inverse function is like a magic trick that "undoes" what the original function does. If we have a function called ( f(x) ) that takes an input ( x ) and gives an output ( y ), the inverse function, written as ( f^{-1}(y) ), takes that output ( y ) and gives us back the original input ( x ). We can also see this relationship using ordered pairs. For example, if ( (a, b) ) is a point on the function ( f ), then ( (b, a) ) is a point on the inverse function ( f^{-1} ).

To dive deeper into notation, if we have a function ( f: A \to B ) that connects a set ( A ) to a set ( B ), then the inverse function ( f^{-1}: B \to A ) takes us back from ( B ) to ( A ). It's important to note that not every function has an inverse. A function can only have an inverse if it is one-to-one, meaning each output comes from exactly one input.

You can check if a function has an inverse by using the Horizontal Line Test. To do this, look at the graph of the function and see if any horizontal line crosses it more than once. If it does, then the function is not one-to-one and doesn’t have an inverse. A function that passes this test can have an inverse.

Now, let’s look at how to find the inverse of a function step by step:

1. Replace: Start with the function written as ( y = f(x) ).
2. Switch: Swap ( x ) and ( y ) to get ( x = f(y) ).
3. Solve: Solve this new equation for ( y ) in terms of ( x ). This gives you the inverse function.
4. Notation: Write the inverse function as ( f^{-1}(x) ) to show that it reverses the original function.

Here’s an example. Let’s say we have the function described by ( y = 2x + 3 ). To find its inverse:

1. Start with ( y = 2x + 3 ).

2. Switch the variables: ( x = 2y + 3 ).

3. Solve for ( y ):

   $$x - 3 = 2y$$ $$y = \frac{x - 3}{2}$$

4. So, the inverse function is ( f^{-1}(x) = \frac{x - 3}{2} ).

It's important to check that when we apply the original function followed by its inverse, we get back to our original input ( x ). This is noted as ( (f \circ f^{-1})(x) = x ) and ( (f^{-1} \circ f)(x) = x ).

Also, when writing the inverse function, remember to think about the domain (inputs) and range (outputs) of both functions. The domain of the original function becomes the range of the inverse, and the range of the original function becomes the domain of the inverse. This shows how outputs of one function link to inputs of the other.

For example, if we look at the function ( f(x) = x^2 ) where ( x \geq 0 ), its inverse is ( f^{-1}(x) = \sqrt{x} ). This works only when ( x \geq 0 ) because of the domain.

In conclusion, writing and understanding inverse functions involves using clear symbols and following steps to reverse what the function does. Knowing that a function needs to be one-to-one, using the Horizontal Line Test, and carefully finding and writing the inverse helps you master this topic. Always check your results with function composition and keep an eye on the domain and range.