The domain of a function is the set of all valid input values, and the range is the set of all possible output values. To find the domain, ask: "What values of x could cause a problem?" Division by zero and square roots of negative numbers are the two main restrictions. For f(x) = 1/(x - 3), the domain is all real numbers except x = 3. For f(x) = √(x + 2), the domain requires x + 2 ≥ 0, so x ≥ -2.
Function composition, written (f ∘ g)(x) = f(g(x)), means applying one function inside another. Given f(x) = x² + 1 and g(x) = 3x, then f(g(x)) = (3x)² + 1 = 9x² + 1. The order matters: g(f(x)) = 3(x² + 1) = 3x² + 3, which is different. Composition chains can extend to three or more functions: f(g(h(x))).
An inverse function reverses the effect of the original function. If f maps input a to output b, then f⁻¹ maps b back to a. To find an inverse algebraically, replace f(x) with y, swap x and y, solve for y, and replace y with f⁻¹(x). For f(x) = 2x + 5: set y = 2x + 5, swap to x = 2y + 5, solve y = (x - 5)/2, so f⁻¹(x) = (x - 5)/2.
Not every function has an inverse. Only one-to-one (injective) functions do — those where each output comes from exactly one input. The horizontal line test determines this: if any horizontal line crosses the graph more than once, the function fails and has no inverse. A function and its inverse are reflections of each other across the line y = x, and their compositions yield the identity: f(f⁻¹(x)) = x.