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.

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 ).

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 ).

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.

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} ).

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} ).

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) ).

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 ).

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.

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} ).

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.

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.

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.

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.

The swap method involves replacing ( f(x) ) with ( y ), swapping ( x ) and ( y ), solving for ( y ), and then replacing ( y ) with ( f^{-1}(x) ).

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.

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.

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.

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.

An inverse function reverses the input-output relationship of the original function. It 'undoes' what the original function does, returning the original input.

Function: Maps x to y | Inverse: Maps y to x

Function: Domain is x-values, Range is y-values | Inverse: Domain is y-values, Range is x-values

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

Function: Original graph | Inverse: Reflection of the original graph across the line ( y = x )

Linear: Straightforward algebraic manipulation | Quadratic: Requires domain restriction to ensure one-to-one property, more complex algebra

( 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

Horizontal Line Test: Checks if a function is one-to-one (invertible) | Vertical Line Test: Checks if a relation is a function

( f(x) = 2x + 3 ): Simple algebraic steps | ( f(x) = sqrt{x} ): Squaring operation needed, domain consideration

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.

( f(x) = sin(x) ): Not invertible without domain restriction | ( f(x) = arcsin(x) ): Invertible by definition, restricted domain