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Monotonicity and parity
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A Functions & Sequences set, untimed
Named traps
7 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Composite monotonicity mis-combinedCombines inner and outer monotonic behaviors incorrectly for a composite function.→
- Constant term breaks symmetryMisapplies the symmetry test to a function with a constant term that destroys oddness.→
- Even called oddLabels an even function odd (or neither) by misapplying f(-x)=-f(x).→
- Increasing/decreasing sign flippedConfuses where f'(x)>0 with where f'(x)<0, reversing monotonic direction.→
- Parity domain symmetry ignoredDeclares parity without checking the domain is symmetric about 0.→
- Quadratic vertex vs intervalIgnores where a quadratic's vertex sits relative to the queried interval.→
- Shift not accounted in parityConcludes f is odd/even without accounting for an additive shift (e.g. f(x)+1 is odd, not f).→
Key formulas
9- Functiony = f(x), domain D, range Requal functions require same domain AND same rule
- Domain conditionsdenominator ≠ 0even-root radicand ≥ 0log argument > 0base of non-integer power > 0intersect all conditions
- Even: f(−x) = f(x) (symmetric about y-axis)Odd: f(−x) = −f(x) (symmetric about origin)parity requires a domain symmetric about 0
- Power functiony = xa (coefficient must be 1)
- Exponentialy = ax (a>0, a≠1)natural base e
- Logarithmy = loga x, inverse of axln = loge, lg = log10
- Log rulesloga(MN)=loga M+loga Nloga(M/N)=loga M−loga Nloga(Mk)=k·loga M
- Change of baseloga b = logc b / logc aidentities aloga N=N, loga(an)=n
- Periodicityf(x + T) = f(x)
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
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