Inverse functions are mathematical opposites, undoing what the original function does. They're like a rewind button for math operations, allowing us to reverse calculations and find unknown inputs. This concept is crucial for solving equations and modeling real-world situations.

Graphically, inverse functions are mirror images of each other across the line y=x. This visual relationship helps us understand how inverses work and their properties. Inverse functions have many practical applications, from finding roots to modeling growth and decay in various fields.

Inverse Functions

Solving for inverse functions

• Inverse functions "undo" each other, meaning if $f(x)$ and $g(x)$ are inverses, then $f(g(x)) = x$ and $g(f(x)) = x$
• To find the inverse of a function $f(x)$, replace $f(x)$ with $y$, swap $x$ and $y$, solve the equation for $y$, and replace $y$ with $f^{-1}(x)$
• For polynomial functions, follow the steps above and solve the resulting equation for $y$ using algebraic methods (factoring, quadratic formula)
• For rational functions (including reciprocal functions), follow the steps above, solve the resulting equation for $y$ using algebraic methods, and be cautious of domain restrictions as they may cause the function to be undefined for certain values

Graphing inverse functions

• The graph of an inverse function is the reflection of the original function across the line $y = x$
• To graph an inverse function, first graph the original function, then reflect the graph across the line $y = x$
• The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function
  • If the original function is not one-to-one (injective function), restrict its domain to make it one-to-one before finding the inverse (horizontal line test)
• Inverse functions have the same symmetry properties as the original function (even, odd, or neither)
• The x-intercepts of the original function become the y-intercepts of the inverse function, and vice versa (roots and zeros)

Applications of inverse functions

• Radical functions are the inverses of power functions, for example, the inverse of $f(x) = x^2$ is $f^{-1}(x) = \sqrt{x}$
• Radical functions can be used to model situations involving square roots, cube roots, or other roots
  • The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$, to find the radius of a sphere given its volume, use the inverse function $r = \sqrt[3]{\frac{3V}{4\pi}}$
• Inverse functions can be used to "undo" a given function and solve for the input value
  • If the height of an object thrown upward is given by $h(t) = -16t^2 + 64t + 5$, use the inverse function to determine the time at which the object reaches a specific height (quadratic equation)

Special Types of Functions and Their Inverses

• Exponential functions and logarithmic functions are inverses of each other, with important applications in various fields such as finance, biology, and physics
• Piecewise functions can have inverses, but special care must be taken to ensure the function is one-to-one over its entire domain before finding the inverse