Algebra and functions

Explanation

• For a non-zero base, $a^m a^n=a^{m+n}$, $a^m/a^n=a^{m-n}$ and $(a^m)^n=a^{mn}$. Interpret rational powers through roots: $a^{p/q}=\sqrt[q]{a^p}=(\sqrt[q]{a})^p$ whenever the real-valued expression is defined.
• For example, $16^{3/4}=(\sqrt[4]{16})^3=2^3=8$.
• A negative exponent means reciprocal, not a negative value: $a^{-r}=1/a^r$; also avoid applying index laws across addition.
• The base must be non-zero for division and negative powers.
• When an equation contains different-looking powers, rewrite them with a common positive base before equating exponents; this makes the index law being used visible and avoids premature decimal approximations.

Explanation

• A surd is an exact irrational root; simplify it by extracting the largest square factor, as in $\sqrt{50}=5\sqrt2$. Combine only like surds, and rationalise a binomial denominator by multiplying numerator and denominator by its conjugate.
• For example, $1/(3+\sqrt2)=(3-\sqrt2)/(9-2)=(3-\sqrt2)/7$.
• Do not split roots across addition: $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$; after squaring an equation, check every candidate in the original.
• Keep surd answers exact unless the question requests a decimal.
• In equations, record the domain before squaring and substitute every candidate back into the original radical equation because squaring can introduce an extraneous solution.

Explanation

• For $ax^2+bx+c=0$ the discriminant $b^2-4ac$ decides the roots: positive for two distinct real roots, zero for a repeated root, negative for none.
• Questions rarely say “discriminant”: “meets the curve at two distinct points” means substitute the line into the curve, collect a quadratic in $x$ equal to zero, and set $b^2-4ac>0$; “does not meet” means $<0$; “touches” means $=0$.
• The resulting inequality in $k$ needs its critical values and the correct region, written with strict signs and “or” where the region is two-sided.
• Completing the square, $x^2-6x+11=(x-3)^2+2$, gives the vertex $(3,2)$, the minimum value and the range, and proves an expression is always positive.
• A quadratic in a function of the unknown, such as $e^{2x}-5e^x+6=0$ or $\tan^2\theta-\tan\theta=0$, is solved by substituting $u$ for the function, solving for $u$, then returning to $x$ and rejecting values the function cannot take.

Explanation

• A simultaneous solution is an ordered pair satisfying every equation; graphically, solutions are intersection points. Use elimination when coefficients align conveniently, or substitute an expression from the linear equation into the nonlinear equation.
• For example, substituting $y=5-x$ into $xy=6$ gives $x(5-x)=6$, whose roots generate the two intersection pairs. After finding one coordinate, substitute each possible value back separately; losing the second quadratic root or mismatching coordinates are common errors.
• If substitution produces a quadratic, its two real roots may give two intersection points.
• Pair each value of the first coordinate with the value obtained from the same substitution, then verify both coordinates in both original equations.
• Equations involving powers such as $y=2^x$ can similarly be reduced by substituting the exponential expression.

Explanation

• An inequality solution is a set of values; endpoints are included for $\leq$ or $\geq$ and excluded for $<$ or $>$. For a quadratic or rational inequality, locate every zero and undefined value, then use a sign diagram or graph on the resulting intervals.
• For example, $(x-2)(x+3)<0$ between its roots, so $-3<x<2$.
• Multiplying by an expression of unknown sign can reverse the inequality unpredictably; bring a fractional inequality to one side and analyse signs instead.
• Mark open or closed endpoints carefully and exclude any value that makes an original denominator zero, even when it appears as a boundary on the sign diagram.
• For a two-variable graphical inequality, draw a solid boundary for $\leq$ or $\geq$, a dashed boundary for $<$ or $>$, test a point and shade the correct region.

Explanation

• Polynomial manipulation uses expansion, collection and factorisation while preserving equality; choose the form that exposes the required structure. The factor theorem states that $ax-b$ is a factor of $f(x)$ exactly when $f(b/a)=0$; after finding a factor, divide by the linear expression to obtain the remaining polynomial.
• For example, $f(2)=0$ for $f(x)=x^3-3x^2-4x+12$, and division by $x-2$ gives $x^2-x-6$. Cancel factors, not separate terms, in rational expressions, and retain exclusions from the original denominator even when a factor cancels.
• Only linear divisors $ax+b$ or $ax-b$ are required for algebraic division.
• When simplifying a rational expression, factor the complete numerator and denominator before cancelling.
• A cancelled factor still records an excluded input from the original expression, so state that restriction with the simplified answer.

Explanation

• A useful sketch records intercepts, turning points, end behaviour, symmetry and asymptotes rather than relying on a table of isolated points.
• For $y=a/x$ the coordinate axes are asymptotes and the relation is inverse proportionality; $y=a/x^2$ is inverse-square, symmetric about the $y$-axis and has the same sign as $a$.
• Translations such as $y=a/(x-p)+q$ move the asymptotes to $x=p$ and $y=q$.
• The graph of $y=|f(x)|$ reflects negative outputs above the $x$-axis; use its intersections with another graph to solve equations or inequalities.
• Do not draw a reciprocal curve crossing an asymptote, and distinguish direct proportionality $y=kx$ from a relation that merely has positive correlation.

Explanation

• For a composite, substitute the whole of the inner function into the outer one and simplify: $\mathrm{gf}(x)$ with $\mathrm g(x)=3\ln x$ and $\mathrm f(x)=e^x$ is $3\ln e^x=3x$, and leaving $3\ln e^x$ is not a finished answer.
• Solving $\mathrm{fg}(x)=k$ means forming the equation, solving it, then rejecting any solution outside the domain of the inner function, with the reason stated.
• For an inverse, write $y=\mathrm f(x)$, make $x$ the subject, then swap the letters and write the answer as $\mathrm f^{-1}(x)=\ldots$; a rational function needs cross-multiplying and collecting the $x$-terms on one side.
• The domain of $\mathrm f^{-1}$ is the range of $\mathrm f$ and must be stated, found from a sketch, the minimum of a quadratic or the endpoint of the domain.
• A many-to-one function such as a quadratic has no inverse unless its domain is restricted; a repeated output, $\mathrm g(0)=\mathrm g(3)$, is the standard reason.

Explanation

• Transformations outside $f$ act on output coordinates: $y=af(x)$ scales vertically by factor $|a|$ and reflects in the $x$-axis when $a<0$. Transformations inside $f$ act oppositely on inputs: $f(x+a)$ moves left by $a$, while $f(ax)$ scales horizontally by factor $1/|a|$ and may reflect in the $y$-axis.
• The graph $y=|f(x)|$ reflects every negative part above the $x$-axis.
• For $y=|f(-x)|$, first reflect $f$ in the $y$-axis, then apply the output modulus.
• For example, a point $(u,v)$ on $y=f(x)$ maps to $(u/2,v+3)$ on $y=f(2x)+3$.
• Map a general point through combined transformations so horizontal scale factors and shifts stay in the correct order.

Explanation

• Factor the denominator first and assign one partial-fraction term to each linear factor. A repeated factor needs a term for every power: $(x-a)^2$ requires $A/(x-a)+B/(x-a)^2$.
• After multiplying through by the common denominator, substitute convenient roots or compare coefficients to determine the constants.
• If the numerator degree is at least the denominator degree, divide first; omitting a repeated-factor term produces an identity that cannot hold.
• Recombine the fractions to check the identity.
• The simpler terms can then be differentiated, integrated or expanded as binomial/geometric series within the required domain.

Explanation

• A function model links clearly defined variables over a stated domain; it may use polynomial, exponential, logarithmic, trigonometric or reciprocal functions.
• Use data to determine parameters, then calculate and interpret the result with units and appropriate rounding.
• For example, $P(t)=P_0r^t$ models repeated growth, while a sine function can model periodic height and a reciprocal function can model inverse variation.
• Evaluate assumptions such as fixed rates, exact periodicity, ignored constraints or extrapolation.
• A refinement must answer the weakness: use a piecewise rate when growth changes, add a seasonal term when residuals are periodic, or restrict the domain when extrapolation is unreliable.