What does it mean for two functions to be inverses?

They undo each other. If one function changes an input, its inverse takes the result back to that input. For instance, imagine a function that takes a number, doubles it, and adds 3. Its inverse must subtract 3 and then divide by 2. The order matters because the inverse reverses the original steps. More formally, for functions f and g to be inverses, applying them in either order must return the starting input: g(f(x)) = x and f(g(x)) = x, wherever those compositions are defined. [S1]

Let f(x) = 2x + 3 and g(x) = (x − 3)/2. Try an input, say 4: f(4) = 11, and then g(11) = (11 − 3)/2 = 4. In reverse, g(4) = 1/2, and f(1/2) = 2(1/2) + 3 = 4. Each route comes back to 4.

The same check works with a general input. First:

1. g(f(x)) = ((2x + 3) − 3)/2 = x.
2. f(g(x)) = 2((x − 3)/2) + 3 = x.

Both compositions simplify to the original input, so these functions are inverses on the real numbers. [S1]

Checking both orders confirms that each function undoes the other, not just that one order happens to return the input. A useful check is to substitute the entire output of the first function into the second; do not simply swap symbols or cancel parts that are not factors.

For the example, g(f(x)) checks “do f, then undo with g,” while f(g(x)) checks “do g, then undo with f.” The two expressions are different calculations, even though both simplify to x here. The inverse-function test uses both compositions, on the relevant domains. [S1]

Not every function has an inverse function on its full domain. A function needs to be one-to-one on the domain being used; if it is not one-to-one there, restricting its domain may make it one-to-one. [S1] For example, a function that gives the same output for two different inputs cannot be undone uniquely from that output alone.

Use this reusable check:

1. Write down f and the proposed inverse g, including the domains you are using.
2. Substitute f(x) into g and simplify g(f(x)).
3. Substitute g(x) into f and simplify f(g(x)) separately.
4. Check that both results are x for inputs where each composition is defined.
5. If a function fails the one-to-one requirement on its chosen domain, consider whether a restricted domain is intended.

For f(x) = 2x + 3 and g(x) = (x − 3)/2, both composition checks give x, so the pair passes. This test establishes the inverse relationship for the stated domains; it does not by itself tell you which domain to choose in a different problem.