Summary of Inverse Functions

EsSential Concepts

  • For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
  • If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
  • For a function f and its inverse f−1,f(f−1(x))=x for all x in the domain of f−1 and f−1(f(x))=x for all x in the domain of f.
  • Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
  • The graph of a function f and its inverse f−1 are symmetric about the line y=x.

Key Equations

  • Inverse functions
    f−1(f(x))=x  for all  x  in  D, and  f(f−1(y))=y  for all  y  in  R.

Glossary

horizontal line test
a function f is one-to-one if and only if every horizontal line intersects the graph of f, at most, once
inverse function
for a function f, the inverse function f−1 satisfies f−1(y)=x if f(x)=y
inverse trigonometric functions
the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions
one-to-one function
a function f is one-to-one if f(x1)≠f(x2) if x1≠x2
restricted domain
a subset of the domain of a function f