Vocabulary
Inverse Relation- A relation that interchanges the input and output values of the original relation. The graph of an inverse relation is a reflection of the graph of the original relation, with y=x as the line of reflection.
Inverse Function- An inverse relation that is a function. Functions f and g are inverses provided that f(g(x))=x and g(f(x))=x.
Inverse Function- An inverse relation that is a function. Functions f and g are inverses provided that f(g(x))=x and g(f(x))=x.
Essential Information
Follow the example to learn how to find an inverse relation:
Inverse Functions
Functions f and g are inverses of each other provided:
f(g(x))=x and g(f(x)=x
The function g is denoted by f^-1, read as "f inverse."
Follow the example to learn how to verify that functions are inverses:
Functions f and g are inverses of each other provided:
f(g(x))=x and g(f(x)=x
The function g is denoted by f^-1, read as "f inverse."
Follow the example to learn how to verify that functions are inverses:
For most real world problems however, you cannot simply exchange domain and range values, you need to solve the equation for the other value instead
Inverses of Nonlinear Functions
For power functions and all functions their respective inverses are reflected across y=x. But if the original equation is a function, that doesn't necessarily mean that its inverse is also a function. Functions with even power exponents have inverses that are not functions, therefore their domains must be restricted to non-negative real numbers, so the inverse will be a function.
For power functions and all functions their respective inverses are reflected across y=x. But if the original equation is a function, that doesn't necessarily mean that its inverse is also a function. Functions with even power exponents have inverses that are not functions, therefore their domains must be restricted to non-negative real numbers, so the inverse will be a function.
Horizontal Line Test
The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once.
The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once.