All Flashcards
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) )
How can you visually determine if a function has an inverse using its graph?
Use the horizontal line test. If any horizontal line intersects the graph more than once, the function does not have an inverse.
What does the reflection of a function's graph across the line ( y = x ) represent?
It represents the graph of the inverse function.
If a function's graph is increasing, what can you say about its inverse's graph?
The inverse function's graph will also be increasing.
How does restricting the domain of a function affect its graph and invertibility?
Restricting the domain can make a function one-to-one, allowing it to have an inverse. The graph will only show the portion of the function within the restricted domain.
Given the graph of ( f(x) ), how do you sketch the graph of ( f^{-1}(x) )?
Reflect the graph of ( f(x) ) across the line ( y = x ).
What does a vertical asymptote on the graph of ( f^{-1}(x) ) indicate about the graph of ( f(x) )?
It indicates a horizontal asymptote on the graph of ( f(x) ).
If the graph of ( f(x) ) passes through (a, b), what point must the graph of ( f^{-1}(x) ) pass through?
The graph of ( f^{-1}(x) ) must pass through (b, a).
How can you identify the domain and range of a function from its graph?
The domain is the set of all x-values covered by the graph, and the range is the set of all y-values covered by the graph.
What does it mean if a function's graph is symmetric about the line ( y = x )?
It means the function is its own inverse, i.e., ( f(x) = f^{-1}(x) ).
How does the slope of a function's graph relate to the slope of its inverse's graph?
The slope of the inverse function at a point is the reciprocal of the slope of the original function at the corresponding point.
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 ).