Functions - Solving Equations and Inequalities

Graphical Solutions to Equations: To solve $f(x) = g(x)$, identify the $x$-coordinates of the intersection points of the two graphs. Visually, this is where the curves $y = f(x)$ and $y = g(x)$ cross each other on the Cartesian plane.

Solving Inequalities Graphically: For $f(x) > g(x)$, look for the intervals of $x$ where the graph of $y = f(x)$ is strictly above the graph of $y = g(x)$. For $f(x) \leq g(x)$, find where the graph of $f$ is below or touching the graph of $g$.

Quadratic Inequalities and Parabolic Shapes: To solve $ax^2 + bx + c > 0$, first find the roots (x-intercepts). If $a > 0$, the graph is a 'U-shaped' parabola opening upwards; the solution is the region outside the roots. If $a < 0$, the graph is an 'n-shaped' parabola opening downwards; the solution is the region between the roots.

Modulus Equations: The equation $|f(x)| = g(x)$ is solved by considering two cases: $f(x) = g(x)$ and $f(x) = -g(x)$. Visually, the modulus function $|f(x)|$ takes any part of the graph of $f(x)$ that is below the $x$-axis and reflects it above the $x$-axis, creating a 'V' or 'W' shape.

Rational Inequalities and Asymptotes: When solving $\frac{P(x)}{Q(x)} \geq 0$, you must identify critical values from both the numerator ($P(x)=0$) and the denominator ($Q(x)=0$). Visually, the zeros of the denominator represent vertical asymptotes where the function is undefined, so these values can never be included in the solution set (always use open brackets).

Domain Restrictions: Many equations involving square roots $\sqrt{f(x)}$, logarithms $\ln(f(x))$, or fractions have implicit restrictions ($f(x) \geq 0$, $f(x) > 0$, or denominator $\neq 0$). Solutions must be checked against these domains to avoid 'extraneous solutions' that appear algebraically but do not exist on the graph.

Inverse Function Intersections: The graphs of $f(x)$ and $f^{-1}(x)$ are reflections of each other across the diagonal line $y = x$. Therefore, solving $f(x) = f^{-1}(x)$ is often equivalent to solving $f(x) = x$, which is usually simpler.