1: Functions and graphs

Functions and Graphs

A function is a fundamental mathematical concept describing a specific relationship between two sets: the domain (set of all possible inputs) and the range (set of all corresponding outputs). Crucially, each input value ($x$) from the domain is assigned to exactly one output value ($y$). This is often expressed as $y = f(x)$, where $f$ denotes the function rule.

Functions are commonly represented graphically on the Cartesian plane. The graph of a function $f$ is the set of all points $(x, y)$ where $y = f(x)$. The Vertical Line Test provides a quick visual check: if any vertical line intersects the graph at more than one point, the graph does not represent a function. This enforces the "one output per input" rule.

When analyzing a function's graph, identify key features:

• Intercepts: Where the graph crosses the axes. The x-intercept(s) ($f(x) = 0$) and y-intercept ($f(0)$).
• Intervals of Change: Sections where the function is increasing (graph rises as $x$ increases), decreasing (graph falls as $x$ increases), or constant.
• Local Extrema: Points where the function changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum).
• Symmetry: Even functions satisfy $f(-x) = f(x)$, exhibiting symmetry about the y-axis (e.g., $f(x) = x^2$). Odd functions satisfy $f(-x) = -f(x)$, exhibiting rotational symmetry about the origin (e.g., $f(x) = x^3$).
• Asymptotes: Lines (vertical, horizontal, or oblique) that the graph approaches infinitely closely but never touches, indicating undefined values or behavior at infinity (common in rational functions).

Recognize basic function types and their characteristic graphs:

• Linear: $f(x) = mx + b$, straight line with slope $m$ and y-intercept $b$.
• Quadratic: $f(x) = ax^2 + bx + c$, parabola opening upwards ($a > 0$) or downwards ($a < 0$).
• Polynomial: Sums of terms $a_nx^n + \dots + a_1x + a_0$. Graphs are smooth, continuous curves; behavior for large $|x|$ depends on the leading term.
• Rational: Quotients of polynomials ($f(x) = P(x)/Q(x)$). Graphs often have discontinuities (holes, vertical asymptotes) where $Q(x) = 0$ and horizontal/oblique asymptotes describing end behavior.
• Root: $f(x) = \sqrt{x}$ (or other roots), typically defined for $x \geq 0$ (if even root), producing characteristic curved shapes starting at the origin or an intercept.