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2.6. DERIVATIVES OF INVERSE FUNCTIONS
(c) Next consider the new function defined by p(x) = F(C(x)). Use the formulas for
F and C to determine an expression for p(x) and simplify this expression as much
as possible. What do you observe?
(d) Now, let r(y) = C(F(y)). Use the formulas for F and C to determine an expression
for r(y) and simplify this expression as much as possible. What do you observe?
(e) What is the value of C ′ (x)? of F ′ (y)? How do these values appear to be related?
⊲⊳
Basic facts about inverse functions
A function f : A → B is a rule that associates each element in the set A to one and only
one element in the set B. We call A the domain of f and B the codomain of f . If there
exists a function g : B → A such that g( f (a)) = a for every possible choice of a in the
set A and f (g(b)) = b for every b in the set B, then we say that g is the inverse of f .
We often use the notation f −1 (read “ f -inverse”) to denote the inverse of f . Perhaps the
most essential thing to observe about the inverse function is that it undoes the work of f .
Indeed, if y = f (x), then
f
−1 (y) = f
−1 ( f (x)) = x,
and this leads us to another key observation: writing y = f (x) and x = f −1 (y) say the
exact same thing. The only difference between the two equations is one of perspective –
one is solved for x, while the other is solved for y.
Here we briefly remind ourselves of some key facts about inverse functions. For a
function f : A → B,
• f has an inverse if and only if f is one-to-one 8 and onto 9 ;
• provided f −1 exists, the domain of f −1 is the codomain of f , and the codomain of
f −1 is the domain of f ;
• f −1 ( f (x)) = x for every x in the domain of f and f ( f −1 (y)) = y for every y in the
codomain of f ;
• y = f (x) if and only if x = f −1 (y).
The last stated fact reveals a special relationship between the graphs of f and f −1 . In
particular, if we consider y = f (x) and a point (x, y) that lies on the graph of f , then it
is also true that x = f −1 (y), which means that the point (y, x) lies on the graph of f −1 .
8 A function f is one-to-one provided that no two distinct inputs lead to the same output.
9 A function f is onto provided that every possible element of the codomain can be realized as an output
of the function for some choice of input from the domain.
2.6. DERIVATIVES OF INVERSE FUNCTIONS
(c) Next consider the new function defined by p(x) = F(C(x)). Use the formulas for
F and C to determine an expression for p(x) and simplify this expression as much
as possible. What do you observe?
(d) Now, let r(y) = C(F(y)). Use the formulas for F and C to determine an expression
for r(y) and simplify this expression as much as possible. What do you observe?
(e) What is the value of C ′ (x)? of F ′ (y)? How do these values appear to be related?
⊲⊳
Basic facts about inverse functions
A function f : A → B is a rule that associates each element in the set A to one and only
one element in the set B. We call A the domain of f and B the codomain of f . If there
exists a function g : B → A such that g( f (a)) = a for every possible choice of a in the
set A and f (g(b)) = b for every b in the set B, then we say that g is the inverse of f .
We often use the notation f −1 (read “ f -inverse”) to denote the inverse of f . Perhaps the
most essential thing to observe about the inverse function is that it undoes the work of f .
Indeed, if y = f (x), then
f
−1 (y) = f
−1 ( f (x)) = x,
and this leads us to another key observation: writing y = f (x) and x = f −1 (y) say the
exact same thing. The only difference between the two equations is one of perspective –
one is solved for x, while the other is solved for y.
Here we briefly remind ourselves of some key facts about inverse functions. For a
function f : A → B,
• f has an inverse if and only if f is one-to-one 8 and onto 9 ;
• provided f −1 exists, the domain of f −1 is the codomain of f , and the codomain of
f −1 is the domain of f ;
• f −1 ( f (x)) = x for every x in the domain of f and f ( f −1 (y)) = y for every y in the
codomain of f ;
• y = f (x) if and only if x = f −1 (y).
The last stated fact reveals a special relationship between the graphs of f and f −1 . In
particular, if we consider y = f (x) and a point (x, y) that lies on the graph of f , then it
is also true that x = f −1 (y), which means that the point (y, x) lies on the graph of f −1 .
8 A function f is one-to-one provided that no two distinct inputs lead to the same output.
9 A function f is onto provided that every possible element of the codomain can be realized as an output
of the function for some choice of input from the domain.