Algebra and Functions
Scholarship Calculus uses Algebra and functions to test structural reasoning: domains, inverse and composite functions, inequalities, and the behaviour of polynomial and rational graphs. Routine manipulation is assumed; marks come from justifying equivalence and handling restrictions precisely.
Core rules and concepts
- A function's domain is part of its definition. Exclude values making a denominator zero, requiring an even root of a negative, or giving a non-positive logarithm argument.
- An inverse exists only when the function is one-to-one on the stated domain. Swap x and y, then solve; the inverse's domain is the original range.
- For inequalities, move all terms to one side and use a sign table. Multiplying by a variable expression is invalid unless its sign is known.
- The factor theorem gives roots; multiplicity controls whether a graph crosses or touches the axis.
- Transformations: f(ax+b) involves a horizontal stretch or compression and a translation; order matters.
Worked example
Solve (x-1)/(x+2) ≥ 2. Subtract 2: (x-1)/(x+2) - 2 ≥ 0, so (x-1-2x-4)/(x+2) = (-x-5)/(x+2) ≥ 0. Multiply by -1 and reverse: (x+5)/(x+2) ≤ 0. Critical values are -5 and -2; x=-2 is excluded. A sign chart gives positive for x<-5, negative between -5 and -2, positive for x>-2. The inequality is satisfied on [-5,-2). Check x=-5: original gives 2≥2, true; x=-3 gives 4≥2, true; x=0 gives -1/2≥2, false.
Traps
Cross-multiplying by x+2 without splitting cases; including excluded denominator zeros; forgetting to reverse an inequality when multiplying by a negative; assuming f^{-1}(x)=1/f(x).