Lecture 21
Calculus of Inverse Functions
Math 13200
Motivating Goal: We would like to understand the “inverse function” of ln x. We will name it
exp x once we have it. First, we need to talk about which functions have inverses.
One-to-one Functions
Definition: A function f (x) is called one-to-one, or injective if for every y-value y0 , there
exists at most one x-value x0 with f (x0 ) = y0 . Another way to write this: if x1 6= x2 , then
f (x1 ) 6= f (x2 ).
On the graph, we can test whether or not a function is one-to-one using the horizontal line test.
Specifically, if there exists a horizontal line y = y0 with more than point on the graph, then the
function is not injective.
Examples
Any strictly monotonic function is injective. Indeed, if x1 6= x2 , then either f (x1 ) < f (x2 ) or
f (x1 ) > f (x2 ) (depending on whether f is increasing or decreasing, and which is bigger x1 or x2 ).
Thus, f (x1 ) 6= f (x2 ), and we conclude that f is one-to-one.
We can see this fact geometrically: Indeed, the horizontal line test can’t hit two point on the graph
of an increasing function. (Otherwise, the function couldn’t have been only incrneasing between
those two points.)
f (x) = 2x f −1 (y) = 21 y
f (x) = x3 − 1 f −1 (y) =
√
3
y+1
1
Inverse of an Injective Function
The horizontal line test is like the vertical line test, but flipping the x and y-axes. So.... what
happens if you flip the x and y axes for an injective function? You get a new function! It is called
the inverse
Specifically, if f (x) is injective, then define f −1 (y) to be the unique value (if it exists) such
that
f (f −1 (y)) = y
Another way of writing this is as:
y = f (x)
⇔
f −1 (y) = x
How do you obtain the inverse function of f (x) from the graph of f (x)?
Well, to evaluate f (x0 ) using the graph of f , you go to x0 on the x-axis, go up to the point on the
graph corresponding to x0 (it is unique by the vertical line test) and the y-coordinate of that point
is f (x0 ). Indeed, that is how the graph is constructed.
So to evaluate f −1 (y0 ), you go to y0 on the y-axis, go to the point on the graph with y-coordinate
y0 (it is unique by the horizontal line test), and the x-coordinate of that point is f −1 (y0 ).
Examples
2
Switching the x and y axes
To study inverse functions, we want to study them in the exact same way that we studied other
functions. Specifically, we want to view them as functions of x with values labeled by y. That’s no
problem - we just switch the x and y!
By doing so, we are reflecting the entire plane in the line y = x.
Key point: The graph of the inverse function f −1 (x) is the reflection of the graph of f (x) in the
line y = x.
Examples
Analyzing what happens to lines under reflection in y = x
We want to study the derivatives of inverse functions. In order to do this, we must understand
what happen to lines under reflection in y = x - since the derivative is the slope of the tangent
line.
Examples
Answer: the line through (c, d) of slope m transforms to the line through (d, c) of slope 1/m.
Inverse Function Theorem
Suppose f is injective. The tangent line of f (x) at x = x0 is the line through (x0 , f (x0 )) with slope
f 0 (x0 ). Since f is injective, we have
(x0 , f (x0 )) = (f −1 (y0 ), y0 )
3
where y0 = f (x0 ). After reflecting the tangent line, we get the line through (y0 , f −1 (y0 )) of slope
1/f 0 (x0 ). We conclude:
Theorem: Suppose f (x) is an injective differentiable function. Then f −1 (y) (where y = f (x)) is
also a differentiable function, and it satisfies
(f −1 )0 (y) =
1
1
= 0 −1
f 0 (x)
f (f (y))
In particular, writing this in terms of x instead of y, f −1 (x) is differentiable, with derivative
(f −1 )0 (x) =
Examples
4
1
f 0 (f −1 (x))