Inverse Functions
In Algebra I, you were most likely taught about operations that had an inverse relationship. You
were taught that to undo addition we use subtraction and to undo multiplication we use
division. We also do these in reverse. It is often helpful to find an inverse for a function, so that
we can see what happens when we ‘undo’ the function’s process.
The first thing we have to consider is whether or not the inverse of a function will also be a
function. Although we can undo most functions, not all of these processes lead to functions,
some lead just to relations. (Remember that a relation is a set of ordered pairs, while a
function is a relation where no element in the domain goes to more than one element in the
range.) Only one to one functions have inverses that are also functions. A one to one function
is a function where no element in the range is used more than once.
To see whether or not a function is one to one, we can use the Horizontal Line Test on its graph.
We do this in the same manner as we do the Vertical Line Test, but (obviously) we use a
horizontal line.
If we have the equation of the function (y in terms of x) we can solve it for x and determine if
that new equation is also a function.
Once we have determined that we do in fact have an inverse function, we can set about finding
its function rule. To do this, follow these steps as they apply:
1. Replace f(x) with y. (If necessary)
2. Exchange x and y. (replace every x with a y and every y with an x)
3. Solve for y.
4. Replace y with f-1(x). (This is read, “f inverse of x”)
That’s all there is to it! Please note that the notation for inverse functions can sometimes be
confused with the reciprocal. It’s all in where the exponent is. If the -1 occurs between the f
and the x, it is most likely the inverse function. To denote the reciprocal, we usually put the
whole f(x) in parentheses and the exponent outside of them. (e.g. (f(x))-1)
So let’s see what that would look like:
Example: Find the inverse of 𝑓(𝑥) =
2𝑥+5
7
.
First, we change the notation a little: 𝑦 =
Then exchange: 𝑥 =
2𝑦+5
7
2𝑥+5
7
Then solve:
7𝑥 = 2𝑦 + 5
7𝑥 − 5 = 2𝑦
7𝑥 − 5
=𝑦
2
𝑦=
7𝑥 − 5
2
And finally:
𝑓 −1 (𝑥) =
7𝑥 − 5
2
To verify that we have the right function, we can compose the function and its inverse. They
should undo each other in both directions. That is: 𝑓(𝑓 −1 (𝑥)) = 𝑥 = 𝑓 −1 (𝑓(𝑥)).
Since we are essentially exchanging the x and y values, the graphs of f(x) and f -1(x) are
reflections of one another across the line y=x. The domain of one becomes the range of the
other. The exception to this rule is when you have to restrict the domain to make the inverse a
function. Consider 𝑓(𝑥) = 𝑥 2 . In order to have a function for its inverse, we must restrict the
domain of the squaring function. We usually restrict it to 𝑥 ≥ 0, which gives us the right side of
the parabola. The inverse for this function is then simply 𝑦 = √𝑥.
We will look at this again when we get to Trigonometry, so keep it handy!
Here are some functions pictured with their inverses:
The cubic and cube root: https://www.desmos.com/calculator/8pr0x8wxpf
The line y=4x+2: https://www.desmos.com/calculator/tqfhl5luqq
The square root and quadratic (restricted domain):
https://www.desmos.com/calculator/evdgfsgzhg