How to Find the Inverse of a Function

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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 g(f(x)) = x.

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 = 7x - 15.

y = 7x - 15 Original function
x = 7y - 15 Replace y with x and x with y.
x + 15 = 7y - 15 + 15 Add 15 to both sides.
x + 15 = 7y Simplify
(x + 15) / 7 = 7y / 7 Divide both sides by 7.
(x + 15) / 7 = y Simplify

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.


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!