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.
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) )
Explain the significance of a function being one-to-one for invertibility.
A one-to-one function ensures that each output has a unique input, allowing for a well-defined inverse function that reverses the mapping without ambiguity.
Explain how to determine if a function is invertible.
A function is invertible if it is one-to-one (passes the horizontal line test) and has an unrestricted domain. This ensures a unique inverse function exists.
Describe the relationship between the graph of a function and its inverse.
The graph of the inverse function is a reflection of the original function across the line ( y = x ). This reflects the swapping of x and y values.
Explain how restricting the domain can make a non-invertible function invertible.
By restricting the domain, we can force the function to be one-to-one over that restricted domain, thus allowing an inverse function to be defined.
Describe the swap method for finding inverse functions.
The swap method involves replacing ( f(x) ) with ( y ), swapping ( x ) and ( y ), solving for ( y ), and then replacing ( y ) with ( f^{-1}(x) ).
What is the importance of checking the domain and range when finding inverse functions?
The domain and range of the original function become the range and domain of the inverse function, respectively. This ensures the inverse function is properly defined.
Explain why not all functions have inverses.
Only one-to-one functions have inverses because each y-value must correspond to a unique x-value. If a function is not one-to-one, its inverse would not be a function.
What is the significance of the line y=x in the context of inverse functions?
The line y=x acts as a 'mirror' where the graph of a function and its inverse are reflections of each other. This visually represents the swapping of x and y values.
Explain how to verify if a found inverse function is correct.
To verify, apply ( f(x) ) and then ( f^{-1}(x) ) to a value. If the result is the original value, the inverse function is likely correct.
Explain the concept of an inverse function as an 'undo' button.
An inverse function reverses the input-output relationship of the original function. It 'undoes' what the original function does, returning the original input.
