All Flashcards
How to determine if ( f(x) = x^2 ) is invertible?
- Graph ( f(x) = x^2 ). 2. Apply the horizontal line test. 3. Since a horizontal line can intersect the graph more than once, ( f(x) = x^2 ) is not invertible unless the domain is restricted.
Steps to find the inverse of ( f(x) = 3x + 2 )?
- Replace ( f(x) ) with ( y ): ( y = 3x + 2 ). 2. Swap ( x ) and ( y ): ( x = 3y + 2 ). 3. Solve for ( y ): ( y = (x - 2) / 3 ). 4. Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = (x - 2) / 3 ).
How to find the domain and range of ( f(x) = sqrt{x - 1} ) and its inverse?
- For ( f(x) ): Domain is ( x geq 1 ), Range is ( y geq 0 ). 2. Find the inverse: ( f^{-1}(x) = x^2 + 1 ). 3. For ( f^{-1}(x) ): Domain is ( x geq 0 ), Range is ( y geq 1 ).
How to verify that ( f^{-1}(x) = (x - 1) / 2 ) is the inverse of ( f(x) = 2x + 1 )?
- Compute ( f(f^{-1}(x)) ): ( f((x - 1) / 2) = 2((x - 1) / 2) + 1 = x - 1 + 1 = x ). 2. Compute ( f^{-1}(f(x)) ): ( f^{-1}(2x + 1) = ((2x + 1) - 1) / 2 = 2x / 2 = x ). 3. Since both compositions equal ( x ), the functions are inverses.
How to find the inverse of ( f(x) = x^3 )?
- Replace ( f(x) ) with ( y ): ( y = x^3 ). 2. Swap ( x ) and ( y ): ( x = y^3 ). 3. Solve for ( y ): ( y = sqrt[3]{x} ). 4. Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = sqrt[3]{x} ).
Given ( f(x) = frac{1}{x} ), find its inverse.
- Replace ( f(x) ) with ( y ): ( y = frac{1}{x} ). 2. Swap ( x ) and ( y ): ( x = frac{1}{y} ). 3. Solve for ( y ): ( y = frac{1}{x} ). 4. Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = frac{1}{x} ).
Find the inverse of ( f(x) = e^x ).
- Replace ( f(x) ) with ( y ): ( y = e^x ). 2. Swap ( x ) and ( y ): ( x = e^y ). 3. Solve for ( y ): ( y = ln(x) ). 4. Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = ln(x) ).
How to determine the domain of the inverse of ( f(x) = sqrt{x+3} )?
- Find the range of ( f(x) ): Range is ( y geq 0 ). 2. The domain of ( f^{-1}(x) ) is the range of ( f(x) ), so the domain of ( f^{-1}(x) ) is ( x geq 0 ).
How do you restrict the domain of ( f(x) = x^2 ) to make it invertible?
- Choose either ( x geq 0 ) or ( x leq 0 ). 2. If ( x geq 0 ), the function is one-to-one and invertible. 3. If ( x leq 0 ), the function is one-to-one and invertible.
If ( f(x) = 5x - 2 ), find ( f^{-1}(7) ).
- Find ( f^{-1}(x) ): ( f^{-1}(x) = frac{x + 2}{5} ). 2. Substitute ( x = 7 ) into ( f^{-1}(x) ): ( f^{-1}(7) = frac{7 + 2}{5} = frac{9}{5} ).
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) )