Functions

Explanation

Functions as rules with stated inputs

A function assigns exactly one output to every input in its domain. Different inputs may share an output, but a single allowed input cannot have two outputs. In f(x) = 2x - 5, the symbol f names the function and f(4) = 3 is the image of 4. The notation f: x maps to 2x - 5 describes the same rule.

The domain is the set of allowed inputs. The range is the set of outputs actually produced from that domain. These sets are part of the function, so the same algebraic rule can define different functions when its domain changes. For example, x squared is not one-one on all real numbers but is one-one on x >= 0.

A relation fails to be a function if an allowed input has more than one image. On a graph, the vertical-line test expresses this requirement: every vertical line must meet the graph at most once. This pack uses real-valued functions unless another domain is stated.

Explanation

Domain restrictions must travel with the rule

For a formula, first identify values that make the expression meaningful. A denominator cannot be zero, and the expression inside a real square root cannot be negative. Thus 1/(x - 3) has domain x != 3, while sqrt(2x + 6) has domain x >= -3.

A question may impose a narrower domain than the formula itself requires. If f(x) = x squared for -2 <= x <= 1, those endpoints determine the attainable outputs. Never silently replace a stated domain by all real numbers, because that can change the range, whether the function is one-one, and whether an inverse exists.

Interval language should distinguish strict and inclusive boundaries. For instance, x > 2 excludes 2, while x >= 2 includes it. A reciprocal may approach zero without attaining it, so its range often has an open boundary even when its domain has a closed endpoint.

Explanation

Finding a range systematically

A range can often be found from a familiar graph, an inequality, or by rearranging y = f(x). For a completed square, use that a real square is non-negative. For a reciprocal, consider its sign, endpoint values and unattained asymptotes. For a stated finite interval, inspect relevant endpoints and turning points.

When rearranging y = f(x), ask which y-values produce an allowed real x. This method exposes restrictions cleanly. If y = sqrt(x + 4) - 2, then y + 2 is a square root and must be non-negative, so y >= -2; conversely x = (y + 2) squared - 4 produces an allowed x for every such y.

Do not infer a range from a few sampled values. Sampling can miss an extremum or an excluded value. Instead, use the structure of the formula together with the entire stated domain, and say whether each boundary is included.

Explanation

Composition means one function acts after another

The composite gf means g after f: gf(x) = g(f(x)). Work from the inside outward. In general fg and gf are different because they apply the rules in opposite orders. Substitute the complete inner expression, using brackets before simplifying.

A composite is defined only when every output under consideration from the inner function is an allowed input for the outer function. In set language, the relevant range of f must lie within the domain of g for gf to be formed on the whole stated domain of f. Otherwise the composite domain must be restricted, if the question permits it.

Formula restrictions remain active after substitution. If g(u) = sqrt(u), then gf(x) requires f(x) >= 0. If g(u) = 1/(u - 2), then gf(x) requires f(x) != 2. Finding the algebraic expression without its domain is incomplete when a restriction arises.

Explanation

One-one functions and domain restriction

A function is one-one when different domain inputs always have different outputs. Equivalently, f(a) = f(b) must force a = b. Graphically, a one-one function passes the horizontal-line test: every horizontal line meets its graph at most once.

A strictly increasing or strictly decreasing function on its stated domain is one-one. A full quadratic normally is not one-one because points equally spaced from its axis have the same output. Restricting the domain to one side of the vertex removes that duplication and can make an inverse possible.

The domain restriction is not cosmetic. The restrictions x >= h and x <= h choose different branches of y = (x - h) squared + k and therefore produce different inverse formulas. Preserve the stated inequality when selecting the sign of a square root.

Explanation

Finding and checking an inverse

Only a one-one function has an inverse function on its stated domain. To find it algebraically, write y = f(x), rearrange to make x the subject, then interchange x and y. Record the inverse domain as the original range and the inverse range as the original domain.

A correct inverse reverses the original mapping. Consequently f inverse(f(x)) = x for every x in the domain of f, and f(f inverse(x)) = x for every x in the domain of f inverse. These identities are strong checks, but each must be used on its valid domain.

The notation f inverse does not mean 1/f(x). A reciprocal changes each output y to 1/y, while an inverse function exchanges input and output roles. For f(x) = 2x + 1, the inverse is (x - 1)/2, not 1/(2x + 1).

Explanation

A function and its inverse on a graph

If f maps a to b, then f inverse maps b back to a. Therefore the point (a, b) on y = f(x) becomes (b, a) on y = f inverse(x). Swapping coordinates is exactly reflection in the line y = x.

The domain and range also swap. A vertical boundary or asymptote x = c for the original corresponds to a horizontal boundary or asymptote y = c for the inverse, and vice versa. Any point on y = x stays fixed under the reflection.

A graphical inverse must still be a function, so the original graph must be one-one. If the original fails the horizontal-line test, its reflection fails the vertical-line test. Restricting the original domain to a one-one branch resolves this when appropriate.

Explanation

Translations: changes outside and inside f

The graph y = f(x) + a is the graph of y = f(x) translated vertically by vector (0, a). Every point (p, q) becomes (p, q + a). A positive a moves the graph up, while a negative a moves it down.

The graph y = f(x + a) is translated horizontally by vector (-a, 0). The sign appears reversed because the same original input p now occurs when x + a = p, so x = p - a. Thus f(x + 4) moves left by 4, not right.

For y = f(x + a) + b, combine the independent moves: translation by vector (-a, b). Track a known point or a vertex to verify the direction instead of relying on a memorised phrase alone.

Explanation

Stretches, reflections and simple combinations

For y = af(x), multiply every y-coordinate by a. If |a| > 1 this is a vertical stretch by factor |a|; if 0 < |a| < 1 it is a vertical compression. A negative a also reflects the graph in the x-axis. A point (p, q) becomes (p, aq). These stretch and reflection rules assume non-zero multipliers; if a = 0 outside the function, the graph collapses to the line y = 0.

For y = f(ax), with a != 0, divide every x-coordinate by a. The horizontal scale factor is 1/|a|. A negative a also reflects in the y-axis. A point (p, q) becomes (p/a, q). This reciprocal horizontal factor is a common source of mistakes, and coordinate division is not defined when a = 0.

For a combination, map coordinates from the original graph or solve the inner argument for the new x-coordinate. In y = A f(B(x - h)) + k, with B != 0, a base point (p, q) maps to (h + p/B, k + Aq). This point rule captures translations, reflections and stretches without ambiguity.

Revision checklist, traps and scope boundary

Language and range: check that each input has one output, carry the stated domain with the rule, and derive the complete range with correct open or closed boundaries. A graph may be a function without being one-one; use vertical and horizontal line tests for the two different questions.

Composition and inverses: read gf from right to left, place the inner rule in brackets, and ensure its range fits the outer domain. Find an inverse only after establishing one-one behaviour; swap domain and range, choose branches from the stated restriction, and check by composition.

Graphs and transformations: inverse graphs reflect in y = x. Outside changes act on y-coordinates; inside changes act reciprocally on x-coordinates. Negative outside and inside factors produce reflections in the x-axis and y-axis respectively. Track a known point to check a combined transformation.

Excluded scope: this pack does not teach quadratics as a separate section, logarithmic or exponential functions, calculus, numerical methods, or any other Paper 1 or A Level component. It covers only the function content explicitly mapped to Pure Mathematics 1 section 1.2.