Functions - Function Concepts (Domain, Range, Inverse, Composite)

Definition of a Function: A relation where every input $x$ in the domain maps to exactly one output $y$ in the range. Visually, a function must pass the Vertical Line Test, meaning any vertical line drawn through the graph intersects the curve at most once.

Domain and Range: The domain is the set of all possible input values ($x$-axis coverage), and the range is the set of all possible output values ($y$-axis coverage). On a graph, you can identify the domain by looking at the left-to-right span and the range by looking at the bottom-to-top span of the curve.

Composite Functions: Formed by substituting one function into another, denoted as $(f \circ g)(x) = f(g(x))$. This represents a sequence of transformations where the output of the inner function $g$ becomes the input for the outer function $f$. For the composite to exist, the range of $g$ must be a subset of the domain of $f$.

Inverse Functions: An inverse function $f^{-1}(x)$ reverses the operation of $f(x)$. A function has an inverse only if it is one-to-one (bijective), which is verified visually using the Horizontal Line Test: a horizontal line should only touch the graph once.

Geometric Properties of Inverses: The graph of an inverse function $f^{-1}(x)$ is a reflection of the original function $f(x)$ across the 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 $f^{-1}$.

Domain Restrictions: Functions that are not naturally one-to-one, such as $f(x) = x^{2}$, require a domain restriction (e.g., $x \geq 0$) to have an inverse. Visually, this involves selecting only one 'branch' of a curve (like the right side of a parabola) so it passes the Horizontal Line Test.

Self-Inverse Functions: Some functions are their own inverse, meaning $f(x) = f^{-1}(x)$. A common visual example is the reciprocal function $f(x) = \frac{1}{x}$, which is symmetrical about the line $y = x$ and $y = -x$.