How to find the inverse of a function

An inverse function, denoted f^{-1}, reverses the effect of f: applying f then f^{-1} returns the original value. The inverse exists only when f is one‑to‑one, which the horizontal line test confirms. When it exists, the domain of f becomes the range of f^{-1} and vice versa, so keeping track of these sets is essential.

Procedure for finding an inverse

Follow these steps to compute the inverse:

1. 1Write the function as y = f(x).
2. 2Solve the equation for x in terms of y.
3. 3Swap the symbols x and y to obtain y = f^{-1}(x).
4. 4State the domain of the inverse, which equals the original range.

Example: let f(x) = 2x + 3. Start with y = 2x + 3. Subtract 3 to get y - 3 = 2x, then divide by 2: x = (y - 3)/2. Finally swap x and y, yielding f^{-1}(x) = (x - 3)/2. The original domain is all real numbers, so the inverse’s range is also all real numbers, and its domain matches the original range.

Key conditions for a function to have an inverse:

• It must be one‑to‑one (passes the horizontal line test).
• Its domain and range must be clearly identified.
• The algebraic manipulation to solve for x must be valid for all allowed values.

Comparison of original function and its inverse for the example:

After finding f^{-1}, verify it by composition: f(f^{-1}(x)) and f^{-1}(f(x)) should both simplify to x on their respective domains. If either composition fails, revisit the algebraic steps or check the one‑to‑one condition. This quick test ensures the inverse is correct before using it in further calculations.