A function gives one output for each allowed input
A function is a rule assigning exactly one output to every input in its domain. The domain is the set of allowed inputs; the range is the set of outputs actually obtained. Writing names the rule, not a multiplication of f by x. Substitution gives f(4) = 11. A relation can associate an input with several outputs, but such a relation is not a function unless its domain or rule is restricted appropriately.
Restrictions can come from algebra or context. For, x = 2 is excluded because the denominator is zero. For, the domain requires x ≥ 3. A model for the number of students cannot ordinarily accept a negative or fractional student count even if its algebraic expression would. State the domain when it matters: an inverse or graph can change when the permitted inputs change.
Read a graph as a collection of input-output pairs
The graph of consists of points (x, f(x)). The vertical-line test asks whether any vertical line meets the graph more than once; if so, the relation is not a function of x on that plotted domain. A function can have several x-intercepts because different inputs may produce zero. Its y-intercept exists when x = 0 is allowed. Distinguish a root, which is an input satisfying f(x) = 0, from the corresponding graph point (x, 0).
For, the x-intercepts are −2 and 2, the y-intercept is −4, and the minimum is (0, −4). Construct a table of values that includes points on both sides of the turning point, choose consistent scales and join smoothly. A cubic such as factors as x(x − 1)(x + 1), with roots −1, 0 and 1. Not every cubic has three distinct real roots. Use the actual equation and its sign changes rather than a single memorised sketch.
Compose functions in the stated order
Composition means using one function's output as another function's input. If and, then (f ∘ g)(x) = f(g(x)) = 2x² + 3. In reverse, (g ∘. These generally differ, so order matters. At x = 1, the first gives 5 and the second 25. The inner function acts first, just as a nested set of brackets is evaluated from inside outward.
The composite domain must permit both stages. If and, then f(g(x)) = 1/(x − 2), excluding x = 2. If and, then g(f(x)) = √(x − 4), requiring x ≥ 4. Simplifying an expression without recording these conditions can incorrectly introduce forbidden inputs. Test a proposed domain by passing an input through the inner and outer rules separately.
Find an inverse only when the mapping permits it
An inverse function reverses a one-to-one mapping. For, solve for x, then exchange the variable names to write f⁻¹(x) = (x − 3)/2. Check and on their permitted domains. The notation f⁻¹ is an inverse function, not 1/f(x). A graph of an inverse is the reflection of the original graph in the line.
The function on all real numbers is not one-to-one because f(2) = f(−2). It has no inverse function on that full domain. Restricting its domain to permits inverse √x with nonnegative range. The domain of a function becomes the range of its inverse, and vice versa. The horizontal-line test helps recognise whether a graph is one-to-one: a horizontal line must not meet it more than once.
Transform a graph carefully
If, then + k shifts the graph k units upward, while y = f(x − h) shifts it h units right. The sign inside the input can be counterintuitive: f(x − 3) uses the old input x − 3, so a known old point at input 0 appears at new x = 3. The graph reflects in the x-axis; reflects in the y-axis. Confirm a transformation by following one specific point.
A vertical scale multiplies all output coordinates by a, while y = f(bx) changes horizontal positions by reciprocal factor 1/b when b is positive. For a in completed-square form, the vertex is (h, k) and the axis of symmetry is x = h. A positive a gives a minimum; a negative a gives a maximum. A graph can solve equations approximately through intersections, but its precision is limited by scale and drawing accuracy.
Define and test a binary operation
A binary operation combines two elements according to a specified rule. If a ★, then 3 ★ 4 = 11, while 4 ★ 3 = 10. Therefore this operation is not commutative. The symbol ★ has no standard arithmetic meaning beyond its definition in the question. A rule is closed on a set if every permitted pair produces an element still in that set. Subtraction is not closed on positive integers because 2 − 5 is not positive.
Commutativity means a ★ ★ a for every allowed pair. Associativity means (a ★ b) ★ ★ (b ★ c); it concerns grouping, not order. An identity e satisfies a ★ ★ a = a for every a. For a ★ on real numbers, e = −2. The inverse of a is a value b satisfying a ★ and b ★ a = e; here. Always check both sides when the operation may not be commutative.
Self-check: for and, find f(g(2)) and f⁻¹(x). The answers are 17 and (x + 1)/3. For a ★, the identity is 0 because a ★ 0 = a = 0 ★ a. Do not infer associativity or distributivity from one numerical example: a general algebraic proof is needed to establish a property for every input, while one valid counterexample is enough to disprove it.
Check a binary operation by algebra
For a ★, both (a ★ b) ★ c and a ★ (b ★ c) simplify to a + b + c + 4, proving associativity on the real numbers. That proof establishes the property for every real triple, unlike checking only one numerical example. A closure claim must also name the set on which the operation is being tested.
Distributivity relates two specified operations
An operation is left-distributive over another when a ★ (b ◇ ★ b) ◇ (a ★ c) for every allowed triple. Right-distributivity has the form (b ◇ c) ★ ★ a) ◇ (c ★ a). Ordinary multiplication distributes over ordinary addition. Addition does not generally distribute over multiplication. The two operations and their order must be named; “the operation is distributive” is incomplete if no second operation is specified.
Consider a ★ and ask whether ordinary multiplication distributes over ★. The left side is a(b ★. The proposed right side is (ab) ★. They differ in general. With a = 2, b = 3, c = 4, the sides are 18 and 16. One counterexample disproves the general property. By contrast, a proof that two symbolic expressions simplify identically establishes a property for every input in the stated domain.
Self-check: test whether subtraction is associative on the integers using 5, 3 and 1. (5 − 3) − 1 = 1, whereas 5 − (3 − 1) = 3, so it is not. Test whether ordinary multiplication distributes over subtraction: for all real a, b and c, so it does. Keep associativity, commutativity and distributivity separate: they ask about grouping one operation, exchanging operands, and relating two operations respectively.