xintercepts

X-intercepts are the points where the graph of a real-valued function y = f(x) crosses the x-axis. In coordinates, these are the points (x, 0) for which f(x) = 0. Such points are also called zeros or roots of the function. An x-intercept exists only if x is a real solution to f(x) = 0 and lies in the domain of f.

To find x-intercepts, solve the equation f(x) = 0. For simple functions this can be done by factoring, applying the quadratic formula, or using numerical methods when a closed-form solution is not available. Graphically, the intercepts are where the graph meets the x-axis. A polynomial of degree n can have up to n real x-intercepts, though some roots may be complex or repeated.

The presence and nature of an intercept around a root depend on its multiplicity. If the corresponding root has odd multiplicity, the graph typically crosses the x-axis there; if even multiplicity, the graph usually just touches and turns around at that point.

Common examples: y = 2x + 3 has an x-intercept at (-3/2, 0). y = x^2 - 4 has intercepts at (-2, 0) and (2, 0). If a function has no real zeros, it has no x-intercepts; if the function is identically zero, every point on the x-axis is an intercept.

Notes: X-intercepts correspond to real roots of f. The y-intercept is a related concept, the point where x = 0, i.e., (0, f(0)).