Functions - Composite and inverse functions

Composite Functions: A composite function, denoted as $(f \circ g)(x)$ or $f(g(x))$, is formed by applying one function to the result of another. Visually, this can be imagined as a 'function machine' chain where the output of $g(x)$ becomes the raw input for $f$. The inner function $g$ is evaluated first, and its result is substituted into the outer function $f$.

Order of Composition: The order in which functions are composed is crucial because composition is generally not commutative. This means $f(g(x))$ is usually not equal to $g(f(x))$. On a mapping diagram, this is seen as following different paths between sets of values, leading to different final destinations.

Inverse Functions: An inverse function, written as $f^{-1}(x)$, performs the opposite operation of the original function $f(x)$. If $f$ maps $x$ to $y$, then $f^{-1}$ maps $y$ back to $x$. Visually, this 'undoes' the transformation, returning the value to its starting point in the domain.

Graphical Relationship of Inverses: The graph of an inverse function $f^{-1}(x)$ is a reflection of the original function $f(x)$ across the identity line $y = x$. If a point $(a, b)$ lies on the graph of $f$, then the point $(b, a)$ must lie on the graph of $f^{-1}$. This creates a symmetrical mirror image relative to the diagonal line passing through the origin.

Existence of an Inverse (One-to-One): A function has an inverse only if it is a 'one-to-one' (injective) function, meaning every $y$-value corresponds to exactly one $x$-value. Visually, you can use the Horizontal Line Test: if any horizontal line crosses the graph of $f(x)$ more than once, the function does not have an inverse over that domain.

Domain and Range Swap: When finding an inverse, the domain of the original function $f$ becomes the range of the inverse function $f^{-1}$, and the range of $f$ becomes the domain of $f^{-1}$. This reciprocal relationship is a direct result of swapping the $x$ and $y$ coordinates.

Identity Property: When a function is composed with its own inverse, they cancel each other out to yield the original input. This is expressed as $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Graphically, the result of this composition is always the straight line $y = x$.