Inverse Functions

Definition and Existence of Inverse Functions

An inverse function reverses the effect of the original function. For a function , the inverse exists if and only if is one-to-one (injective).

• One-to-one function: Each element of the range is mapped from exactly one element of the domain.

• If , then .

• If is not one-to-one, cannot be defined for all in the range.

Example: If maps two different elements in to the same element in , then is not a function.

Inverse of Quadratic Functions

Domain and Range Considerations

Consider with domain and range (all non-negative real numbers).

• is not one-to-one because and .

• The inverse can only be defined on one branch (e.g., ) of the parabola.

Graphical Representation: The graph of is a parabola. Its inverse, , is only defined for and .

Inverse Trigonometric Functions

General Properties

Inverse trigonometric functions are defined by restricting the domain of the original trigonometric functions to intervals where they are one-to-one.

• Notation: , ,

• Important:

Inverse Sine Function ()

• Domain:

• Range:

Example: Find Let . Then with .

Inverse Cosine Function ()

• Domain:

• Range:

Example: Find Let . Then with .

Inverse Tangent Function ()

• Domain:

• Range:

Example: Find Let . Then with .

Evaluating Compositions and Domain Issues

Compositions of Inverse Trigonometric Functions

• For , the result is only if is in .

• For , the result is only if is in .

Example: Since is not in , we use the identity to find the equivalent angle in the principal range.

Undefined Values

• Inverse trigonometric functions are not defined for values outside their domains.

• For example, is not defined because is not in .

• Similarly, is not defined because is not in .

Summary Table: Inverse Trigonometric Functions

Key Points

• Inverse functions exist only for one-to-one functions.

• Quadratic and trigonometric functions require domain restrictions to define their inverses.

• Inverse trigonometric functions have specific domains and ranges to ensure they are functions.

• Compositions like and return only within the principal value ranges.

• Values outside the domain of the inverse trigonometric functions are not defined.