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​ = 7​x​ − 15.

y = 7x - 15 \quad \text{(Original function)} \\ \,\\ x = 7y - 15 \quad \text{(Replace y with x and x with y)}\\ \,\\ x + 15 = 7y - 15 + 15 \quad \text{(Add 15 to both sides.)} \\ \,\\ x + 15 = 7y \quad \text{(Simplify)} \\ \,\\ \frac{x + 15}{7} = \frac{7y}{7} \quad\text{(Divide both sides by 7.)} \\ \,\\ \frac{x + 15}{7} = y \quad\text{(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.

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!