Glossary for Lesson 28: Inverses

Inverse Function

We can think of the inverse function of a function f as the function, denoted f^{-1} that does the opposite operations that f does.  In other words, f^{-1} undoes what f does to the independent variable (usually x ). 

More formally, a function g is the inverse of a function f if f(g(x)) = x for all values in the domain of g and g(f(x)) = x for all values in the domain of f .  If all of that is true, then g(x) = f^{-1}(x) , and f(x) = g^{-1}(x)

We can find the inverse of a function f graphically by reflecting the graph of f in the line y=x .  This is because we can find the inverse of a function by switching its x- and y-coordinates. 

Invertible Function

We call a function an invertible function if it has an inverse function. We can figure out if a function is invertible by performing the horizontal line test or by reflecting its graph in the line y=x and then seeing if this reflection passes the vertical line test.

Horizontal Line Test

In order to figure out if a function has an inverse, we can perform the horizontal line test on that function.  This is very similar to the vertical line test, except that it uses a horizontal line instead. 

To perform the horizontal line test, first graph the function you’re interested in learning about.  Then position something straight, like a ruler or a pencil, parallel to the x -axis on the coordinate plane, and move this “horizontal line” up and down along the y -axis to see if it intersects the graph of the function more than once at any given position.  If it does, then the function is not invertible (i.e. does not have an inverse function). The function y=3x-2 passes the horizontal line test, so it has an inverse. 


The function y=x^3-2x^2-8x does not pass the horizontal line test, as we can see from the graph below, so it is not invertible.