How do I find an inverse function from a simple formula?

Treat the inverse as a new rule that starts with the original output and returns the original input. A reliable procedure is to write y=f(x), solve for x in terms of y, then interchange x and y and write the result as f⁻¹(x). This method requires the function to be one-to-one, or its domain to be restricted so that it is one-to-one. [S1]

For your example:

1. Start with y=2x+3.
2. Solve for x: subtract 3, giving y−3=2x; divide by 2, giving x=(y−3)/2.
3. Swap the variable names: y=(x−3)/2.
4. Write the inverse: f⁻¹(x)=(x−3)/2.

The swap is a relabeling after solving; it does not mean you leave the answer written as x=.... The inverse’s input is the original output, and its output is the original input.

The original equation uses x for an input and y for its output. After solving for x, you have described the original input in terms of the original output. To write that relationship in the usual input-output form for a new function, swap the variable names: the inverse’s input gets called x, and its output gets called y. That is why the formula becomes y=(x−3)/2.

Here, f⁻¹(x) labels the inverse found by reversing the input-output relationship; it is not being used as a reciprocal. For this example, the inverse takes a value, subtracts 3, then divides by 2. Keeping the roles in mind is more useful than memorizing the swap alone: ask, “What original input produced this value?”

Compose the functions in both orders. A correct inverse should undo the original function in either order, wherever the compositions are defined. [S1]

Let f(x)=2x+3 and g(x)=(x−3)/2. Then:

1. Apply f and then g: g(f(x))=((2x+3)−3)/2=x.
2. Apply g and then f: f(g(x))=2((x−3)/2)+3=x−3+3=x.

Both compositions return the starting input, so g is the inverse of f in this example. If your simplification does not return x, check the algebra and the parentheses. For instance, the entire quantity x−3 must be divided by 2.

Use the same solve-and-swap steps when you can isolate the original input as a function of the original output. Before calling the result an inverse function, check the one-to-one condition: if different inputs can give the same output, the function may need a restricted domain before an inverse function is available. [S1] The method alone does not specify an appropriate restriction for every formula, so that depends on the function and the domain in your problem.

Copyable working template:

1. Original rule: y=f(x)= [formula].
2. Solve for the original input: x= [expression in y].
3. Swap the variable names: y= [same relationship written in x].
4. Name the result: f⁻¹(x)= [formula].
5. Check: f(f⁻¹(x))=x and f⁻¹(f(x))=x, where defined.

For 2x+3, line 2 is x=(y−3)/2 and line 4 is f⁻¹(x)=(x−3)/2. Keep the subtraction grouped when dividing; that makes the intended order of operations clear.