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Concept · Functions & Sequences
Piecewise and inverse functions
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A Functions & Sequences set, untimed
Named traps
5 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.
- Branch boundary uncheckedDoes not check the boundary condition between branches, giving an extraneous solution.→
- Inverse domain misstatedStates the wrong domain for the inverse function.→
- Inverse wrong square-root branchChooses the wrong sign of the square-root branch when inverting a restricted quadratic.→
- Reflected-branch zeros missedMisses zeros coming from the negative-x (reflected) branch of an odd piecewise function.→
- Wrong piecewise branchEvaluates using the branch that does not apply to the given input.→
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.
FoundationsPrecalculus bridgeAlgebra 2 foundations