Inverse Functions Rules to Know for Algebra 2

Inverse functions are all about reversing the original function's actions. They take outputs back to inputs, swapping domains and ranges. Understanding how to find and work with these functions is key in Algebra 2, especially when graphing and solving equations.

1. The inverse function reverses the original function.

   • If f(x) takes an input x and produces an output y, then f^(-1)(y) takes y back to x.
   • Inverse functions essentially "undo" the action of the original function.
   • For example, if f(x) = 2x, then f^(-1)(x) = x/2.

2. The domain of a function becomes the range of its inverse, and vice versa.

   • The set of all possible input values (domain) for f(x) becomes the set of output values (range) for f^(-1)(x).
   • Conversely, the output values of f(x) become the input values for f^(-1)(x).
   • Understanding this relationship is crucial for determining the inverse.

3. The notation for an inverse function is f^(-1)(x).

   • The notation f^(-1)(x) does not mean "1 divided by f(x)".
   • It specifically denotes the inverse function of f.
   • This notation helps differentiate between the original function and its inverse.

4. To find the inverse, replace f(x) with y, swap x and y, then solve for y.

   • Start with the equation y = f(x).
   • Swap the variables to get x = f(y).
   • Solve for y to express it as f^(-1)(x).

5. The graphs of a function and its inverse are reflections over the line y = x.

   • If you plot f(x) and f^(-1)(x), they will mirror each other across the line y = x.
   • This reflection property helps visualize the relationship between a function and its inverse.
   • Points on the graph of f(x) will have their coordinates switched on the graph of f^(-1)(x).

6. A function must be one-to-one (injective) to have an inverse function.

   • A one-to-one function means that each output is produced by exactly one input.
   • If a function is not one-to-one, it cannot have an inverse that is also a function.
   • The horizontal line test can be used to determine if a function is one-to-one.

7. The composition of a function and its inverse yields the identity function: f(f^(-1)(x)) = x.

   • This means that applying the function and then its inverse returns the original input.
   • It confirms that f^(-1)(x) is indeed the inverse of f(x).
   • The identity function simply returns the input value unchanged.

8. Not all functions have inverses; some require restricting the domain.

   • Functions that are not one-to-one may need a limited domain to have an inverse.
   • For example, the function f(x) = x^2 is not one-to-one over all real numbers but can be restricted to x ≥ 0.
   • Restricting the domain ensures that each output corresponds to only one input.

9. Inverse trigonometric functions have specific domains and ranges.

   • Inverse trigonometric functions, like arcsin, arccos, and arctan, have limited domains to ensure they are one-to-one.
   • For example, arcsin(x) has a range of [-π/2, π/2].
   • Understanding these domains and ranges is essential for working with inverse trigonometric functions.

10. The derivative of an inverse function is the reciprocal of the original function's derivative.

• If f'(x) is the derivative of f(x), then the derivative of f^(-1)(x) is 1/f'(f^(-1)(x)).
• This relationship is useful for finding slopes of tangent lines to the inverse function.
• It highlights the connection between a function and its inverse in calculus.