All Flashcards
What is a one-to-one function?
A function where each output (y-value) corresponds to only one input (x-value).
What is an invertible function?
A function that has an inverse function. It must be one-to-one and have an unrestricted domain.
What is the inverse function notation for f(x)?
The inverse of a function ( f(x) ) is written as ( f^{-1}(x) ).
Define the domain of a function.
The set of all possible input values (x-values) for which the function is defined.
Define the range of a function.
The set of all possible output values (y-values) that the function can produce.
What is the horizontal line test?
A test to determine if a function is one-to-one. If any horizontal line intersects the graph more than once, the function is not one-to-one.
What does it mean for a domain to be unrestricted?
The function's domain isn't limited in a way that prevents it from having an inverse; it covers all possible input values.
What happens to input-output pairs in an inverse function?
If ( f(a) = b ), then ( f^{-1}(b) = a ). The input and output pairs are flipped.
What is a multivalued inverse?
If a function isn't one-to-one, its inverse might not be a function (it could have multiple outputs for one input).
What is the graphical relationship between a function and its inverse?
The graph of ( f^{-1}(x) ) is a reflection of ( f(x) ) across the line ( y = x ).
How do you denote the inverse of a function f(x)?
( f^{-1}(x) )
If f(a) = b, what is f⁻¹(b)?
( f^{-1}(b) = a )
Given ( y = f(x) ), how do you start finding the inverse?
Swap ( x ) and ( y ) to get ( x = f(y) ).
What is the relationship between the domain of f(x) and the range of f⁻¹(x)?
Domain of ( f(x) ) = Range of ( f^{-1}(x) )
What is the relationship between the range of f(x) and the domain of f⁻¹(x)?
Range of ( f(x) ) = Domain of ( f^{-1}(x) )
What are the differences between a function and its inverse?
Function: Maps x to y | Inverse: Maps y to x
What are the key differences between the domain/range of a function and its inverse?
Function: Domain is x-values, Range is y-values | Inverse: Domain is y-values, Range is x-values
What is the difference between a one-to-one function and a function that is not one-to-one?
One-to-one: Each y-value corresponds to one x-value, invertible | Not one-to-one: Some y-values correspond to multiple x-values, not invertible without domain restriction
Compare the graphs of a function and its inverse.
Function: Original graph | Inverse: Reflection of the original graph across the line ( y = x )
Compare finding the inverse of a linear function vs. a quadratic function.
Linear: Straightforward algebraic manipulation | Quadratic: Requires domain restriction to ensure one-to-one property, more complex algebra
Compare the domain restrictions needed for ( f(x) = x^2 ) vs. ( f(x) = x^3 ) to find inverses.
( f(x) = x^2 ): Requires restriction to ( x geq 0 ) or ( x leq 0 ) | ( f(x) = x^3 ): No restriction needed, already one-to-one
What is the difference between the horizontal line test and the vertical line test?
Horizontal Line Test: Checks if a function is one-to-one (invertible) | Vertical Line Test: Checks if a relation is a function
Compare the steps to find the inverse of ( f(x) = 2x + 3 ) and ( f(x) = sqrt{x} ).
( f(x) = 2x + 3 ): Simple algebraic steps | ( f(x) = sqrt{x} ): Squaring operation needed, domain consideration
What is the difference between ( f(f^{-1}(x)) ) and ( f^{-1}(f(x)) ) when ( f ) is invertible?
Both are equal to x, but ( f(f^{-1}(x)) ) means applying the inverse first, then the function, while ( f^{-1}(f(x)) ) means applying the function first, then the inverse.
Compare the invertibility of ( f(x) = sin(x) ) and ( f(x) = arcsin(x) ).
( f(x) = sin(x) ): Not invertible without domain restriction | ( f(x) = arcsin(x) ): Invertible by definition, restricted domain