All Flashcards
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 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
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) )