Algebra

Explanation

• The commutative laws permit the order to change in addition and multiplication: $a+b=b+a$ and $ab=ba$. The associative laws permit regrouping: $(a+b)+c=a+(b+c)$ and $(ab)c=a(bc)$.
• The distributive law connects multiplication with addition or subtraction: $a(b+c)=ab+ac$ and $a(b-c)=ab-ac$.
• These laws justify rearranging, expanding, factorising and simplifying algebraic expressions.
• Subtraction and division are neither commutative nor associative, so their order cannot be changed in the same way.
• Examiners expect each transformation to preserve equality; a useful check is to identify the law that permits a common factor to be taken out or terms to be reordered.

Explanation

• A function assigns exactly one output to each permitted input. In the notation $f(x)$, $x$ is a placeholder for the input and $f(3)$ means that $3$ replaces every occurrence of $x$ in the function rule.
• A negative or algebraic input should be enclosed in brackets before simplifying.
• The equation $f(x)=k$ asks for input values whose output is $k$; it does not mean $f$ multiplied by $x$.
• Different inputs may share one output, but one input cannot have two outputs if the relation is a function.
• Examiners expect accurate substitution, complete expansion of bracketed inputs and a distinction between evaluating a function and solving an equation involving it.

Explanation

• The domain is the set of permitted input values; the range is the set of output values actually produced. For a finite domain, evaluate every input and list distinct outputs.
• For a continuous restricted domain, consider both endpoints and any turning point within the interval before stating the range.
• Domain restrictions can arise because a denominator cannot be zero or an even square root cannot contain a negative value.
• The domain of a function and its range must be expressed with the correct variable and inequality direction.
• Examiners expect endpoint inclusion to match symbols such as $\leq$ or $<$, and a graph or algebraic argument that identifies the true maximum and minimum outputs.

Explanation

• A composite function applies functions in succession. In AQA notation, $fg(x)$ means $f(g(x))$: $g$ acts first and its output becomes the input to $f$.
• To form an algebraic composite, substitute the entire inner expression into every occurrence of the variable in the outer function, using brackets where needed.
• In general $fg(x)\ne gf(x)$, because reversing the order changes the intermediate value.
• Domain restrictions from both stages must be respected when they matter.
• Examiners expect the action order to be stated or made visible in the substitution; multiplying the two function formulae together does not form a composite function.

Explanation

• An inverse function reverses the action of the original function, so $f^{-1}(f(x))=x$ for inputs in the chosen domain. To find an inverse, write $y=f(x)$, rearrange to make $x$ the subject, then exchange the variable labels.
• The domain of $f^{-1}$ is the range of $f$, and its range is the domain of $f$.
• A function must be one-one on its domain to have an inverse function; a quadratic therefore needs a restriction to one side of its turning point.
• Graphs of a function and its inverse are reflections in $y=x$.
• Examiners expect the correct square-root branch to follow from any stated domain restriction.

Explanation

• Expanding brackets applies the distributive law: every term in one bracket multiplies every term in the other. Products should be written with their signs before like terms are collected.
• Terms are like only when their variable parts and powers match, so $x^2$ and $x$ cannot be combined.
• When an entire expanded expression is subtracted, every sign in that expression changes.
• For three brackets, multiply two brackets first, simplify the result, then multiply by the remaining bracket.
• Examiners expect a fully expanded and collected polynomial in descending powers unless another form is requested; omitted cross-products and mishandled negative signs are the usual sources of lost accuracy.

Explanation

• Row $n$ of Pascal’s triangle gives the coefficients in the expansion of $(a+b)^n$.
• Each interior entry is the sum of the two entries above it.
• Across the expansion, the power of the first term decreases from $n$ to $0$, while the power of the second increases from $0$ to $n$; the two powers in every term add to $n$.
• Numerical coefficients and negative signs inside either term must also be raised to the indicated powers.
• Examiners may ask for a full expansion or one coefficient, so identifying which powers produce the requested variable term can avoid unnecessary expansion.

Explanation

• Factorising reverses expansion by writing an expression as a product. First remove the greatest common numerical and algebraic factor.
• A quadratic can then be factorised by choosing binomial factors whose leading and constant products are correct and whose cross-terms produce the middle coefficient.
• The identity $a^2-b^2=(a-b)(a+b)$ handles a difference of two squares.
• Expressions such as quadratics in $x^2$ may need more than one stage. ‘Factorise fully’ means continue until no factor can be factorised further over the required number set.
• Examiners expect a product, and expansion of the final answer is an efficient check that every original term and sign returns.

Explanation

• Rational expressions are algebraic fractions. Factor numerators and denominators before cancelling, because only common factors—not individual terms—may cancel.
• Addition or subtraction requires a common denominator, with brackets protecting each adjusted numerator. Multiplication combines numerators and denominators after any valid factor cancellation.
• Division means multiplying by the reciprocal of the divisor; the divisor must also be non-zero. Values that make any original denominator zero remain excluded even if a factor later cancels.
• Examiners expect one fully simplified fraction and may require stated restrictions.
• A reliable check is to factor first, identify exclusions from the original expression, then perform the requested operation.

Explanation

• A formula expresses one quantity in terms of others. Substitution should use consistent units and bracket negative values.
• To change the subject, perform inverse operations on both sides while preserving equality. Fractions may first need clearing by multiplication.
• If the new subject appears in more than one term, collect those terms on one side, factor out the subject, then divide by the remaining factor.
• Removing a square can introduce positive and negative roots unless a stated domain or physical context fixes the sign.
• Examiners expect a final expression with the requested subject alone and no occurrence of it elsewhere; substituting the rearranged result back into the original formula can verify equivalence.

Explanation

• The factor theorem states that $x-a$ is a factor of a polynomial $f(x)$ exactly when $f(a)=0$. For a proposed factor $px-q$, the corresponding root is $x=\frac qp$, so rational substitutions must be handled exactly.
• Substitution can verify a factor or determine an unknown coefficient.
• Once one factor is established, polynomial division or coefficient comparison produces the remaining factor.
• A zero remainder proves the proposed factor, but a request to factorise fully or solve the polynomial requires further work.
• Examiners expect the substituted value, the evaluation to zero and an explicit conclusion invoking the factor theorem; merely stating that a factor ‘works’ is insufficient.

Explanation

• Completing the square rewrites a quadratic in the form $a(x-h)^2+k$, exposing its turning point $(h,k)$ and supporting equation solving.
• For $x^2+bx+c$, halve the coefficient $b$ inside the bracket, then compensate outside: $(x+\frac b2)^2+c-\frac{b^2}{4}$.
• If the coefficient of $x^2$ is not $1$, first factor it from the quadratic and linear terms.
• Any compensation inside the bracket is multiplied by that outside factor.
• Examiners expect an identity that expands back to the original expression; the signs in the bracket also reveal the opposite-signed $x$-coordinate of the turning point.

Explanation

• A function sketch must show its defining shape and important features: intercepts, turning points, asymptotes and any restricted domains. Linear graphs are straight; quadratic graphs have a single turning point; exponential graphs $y=ab^x$ and $y=ab^{-x}$ pass through $(0,a)$ when defined.
• For positive $a$ and $b>1$, $ab^x$ increases and $ab^{-x}$ decreases, while both remain positive and approach the horizontal asymptote $y=0$ in one direction.
• A function may be restricted to no more than three domains.
• Included endpoints use filled points and excluded endpoints use open points.
• Examiners expect labelled features and correct endpoint types, not a scale-perfect plot.

Explanation

• A linear equation is solved by applying inverse operations to both sides until the unknown is isolated. A quadratic should first be rearranged into $ax^2+bx+c=0$.
• It may then be solved by factorisation, graph, completing the square or the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
• Factorisation is efficient when integer or simple rational factors exist; the formula works generally and may produce exact surds.
• A graphical solution is read from relevant intersections and is approximate unless exact points are evident.
• Examiners expect all real solutions and method appropriate to the requested accuracy, with the entire formula numerator divided by $2a$.

Explanation

• A simultaneous solution is an ordered pair satisfying both equations; graphically, it is an intersection point. Two linear equations can be solved by elimination or substitution and usually have one intersection unless they are parallel or identical.
• When one equation is second order, substitution normally produces a quadratic, so zero, one or two real intersection points are possible.
• Each resulting $x$-value must be substituted back to find its paired $y$-value.
• Algebraic answers can be checked in both original equations.
• Examiners expect complete coordinate pairs and, in a graphical method, intersections read to the stated accuracy rather than separate unpaired values.