(a) find the inverse function of f, (b) graph both f and...

(a) find the inverse function of $f$, (b) graph both $f$ and $f^{-1}$ on the same set of coordinate
axes, (c) describe the relationship between the graphs of $f$ and $f^{-1},$ and (d) state the domains and ranges of $f$ and $f^{-1}$.
$$f(x)=-\frac{2}{x}$$

Key Concepts

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible outputs. For an inverse function, these roles are interchanged—the domain of the inverse is the range of the original function, and the range of the inverse is the domain of the original. Understanding these sets is crucial when defining functions and ensuring their inverses exist.

Algebraic Manipulation for Solving Equations

Finding the inverse function relies on algebraic techniques, such as swapping variables and solving equations. This process focuses on isolating the new dependent variable, ensuring all occurrences are properly managed so that the new expression correctly represents the inverse function.

Inverse Function

An inverse function essentially reverses the effect of the original function. When finding an inverse, one interchanges the roles of the dependent and independent variables and then solves for the new dependent variable. This process essentially reverses the steps used in the function so that applying the function and its inverse in succession returns the original input.

Graphical Reflection over the Line y = x

The graphs of a function and its inverse are mirror images of each other across the line y = x. This symmetry arises because swapping the roles of x and y reflects every point on the original function across this line, providing a visual confirmation of the inverse relationship.

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