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How to determine if ( f(x) = x^2 ) is invertible?
1. 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 )?
1. 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?
1. 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 )?
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 )?
1. 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.
1. 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 ).
1. 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} )?
1. 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?
1. 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) ).
1. 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 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.