To find an inverse function in math, you must first have a function. It can be almost any set of operations for the independent variable *x* that yields a value for the dependent variable *y*. In general, to determine the inverse of a function of *x*, substitute *y* for *x* and *x* for *y* in the function, then solve for *x*.

#### TL;DR (Too Long; Didn't Read)

In general, to find the inverse of a function of *x*, substitute *y* for *x* and *x* for *y* in the function, then solve for *x*.

## Inverse Function Defined

The mathematical definition of a function is a relation (*x*, *y*) for which only one value of *y* exists for any value of *x*. For example, when the value of *x* is 3, the relation is a function if *y* has only one value, such as 10. The inverse of a function takes the *y* values of the original function as its own *x* values, and produces *y* values that are the original function’s *x* values. For example, if the original function returned the *y* values 1, 3 and 10 when its *x* variable had the values 0, 1 and 2, the inverse function would return *y* values 0, 1 and 2 when its *x* variable had the values 1, 3 and 10. Essentially, an inverse function swaps the *x* and *y* values of the original. In mathematical language, if the original function is f(*x*) and the inverse is g(*x*), then

## Algebra Approach for Inverse Function

To find the inverse of a function involving the two variables, *x* and *y*, replace the *x* terms with *y* and the *y* terms with *x*, and solve for *x*. As an example, take the linear equation, *y* = 7*x* − 15.

The function, (*x* + 15) / 7 = *y* is the inverse of the original.

## Inverse Trigonometric Functions

To find the inverse of a trigonometric function, it pays to know about all the trig functions and their inverses. For example, if you want to find the inverse of *y* = sin(*x*), you need to know that the inverse of the sine function is the arcsine function; no simple algebra will get you there without arcsin(*x*). The other trig functions, cosine, tangent, cosecant, secant and cotangent, have the inverse functions arccosine, arctangent, arccosecant, arcsecant and arccotangent, respectively. For example, the inverse of *y* = cos(*x*) is *y* = arccos(*x*).

## Graph of Function and Inverse

The graph of a function and its inverse is interesting. When you plot the two curves, then draw a line corresponding to the function, *y* = *x*, you’ll notice that the line appears as a “mirror.” Any curve or line below *y* = *x* is “reflected” symmetrically above it. This is true for any function, whether polynomial, trigonometric, exponential or linear. Using this principle, you can graphically illustrate the inverse of a function by graphing the original function, drawing the line at *y* = *x*, then drawing the curves or lines needed to create a “mirror image” that has *y* = *x* as an axis of symmetry.

References

About the Author

Chicago native John Papiewski has a physics degree and has been writing since 1991. He has contributed to "Foresight Update," a nanotechnology newsletter from the Foresight Institute. He also contributed to the book, "Nanotechnology: Molecular Speculations on Global Abundance." Please, no workplace calls/emails!