What Is An Inverse Function?

In mathematics, a function is a process you apply to an independent variable ​x​ to get the dependent variable ​y​. If you think of it as "going from" your ​x​ to arrive at your ​y​, an inverse function goes the opposite way, from the result back to the original value. In a sense, an inverse function is the opposite of the original, "undoing" the process.

Mathematicians define a function as a process or rule that generates the ordered pairs of a set. You can think of the first member of the pair as the ​x​ of the function, and the second member as the ​y​. In a true function, the first value has only one solution value that goes with it. So each ​x​ value has only one corresponding ​y​ value. So, the equation for the horizontal line, ​y​ = 1 is a function, but the vertical line, ​x​ = 1 is not.

The graph of a function and its inverse are reflections of one other, with a line representing ​y​ = ​x​ acting as the "mirror." To take an example, the graph of the natural logarithm function, ln(​x​), starts at negative infinity at the ​y​ axis and just to the right of zero on the ​x​ axis. From there, it crosses the ​x​ axis at the point, (1,0) and has a slightly upward-rising curve over the ​x​ axis. Its inverse, the natural exponent function exp(​x​), has the ​x​-axis as its asymptote, starting at negative infinity on the ​x​ axis, just above it. It crosses the ​y​ axis at (0,1) and curves strongly upward. Draw the two functions on a graph, then draw the line ​y​ = ​x​, and you'll see that exp(​x​) and ln(​x​) mirror each other.

Although the sine and cosine functions are related, one is not the inverse of the other. The sine and cosine functions produce similar graphical results, though cosine "leads" sine by 90 degrees. Also, the cosine is the derivative of the sine. However, the inverse of the sine function is the arcsine, and the inverse of the cosine is the arccosine.

It is relatively easy to find the inverse of many functions: Swap the "​y​" and "​x​" in the equation, and then solve for ​y​. For example, consider the equation

\(y = 2x + 4\)

Swapping y for ​x​ gives

\(x = 2y + 4\)

Subtract 4 from both sides to get

\(x – 4 = 2y\)

and then divide both sides by 2 to get

\(\frac{x}{2} – 2 = y\)

which is the inverse function.

Not all inverses of functions are also functions. Recall that the definition of functions says that every ​x​ has only one ​y​ value. Although arcsine is the inverse of the sine function, arcsine is not technically a function, as ​x​ values have infinitely many corresponding ​y​ values. It's also true with

\(y = x^2 \text{ and } y = \sqrt{x}\)

the first is a function, and the second is its inverse, but the square root gives two corresponding ​y​ values, positive and negative, making it not a true function.