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) )
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 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} ).