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Concept · Functions & Sequences
Domain, range, equal functions, and recovering f(x)
Practice this concept→
A Functions & Sequences set, untimed
Named traps
9 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.
- Denominator zero allowedFails to exclude values that make a denominator zero.→
- Domain conditions not intersectedLists separate conditions but fails to intersect them for the final domain.→
- Equal-function domain ignoredCalls two expressions the same function despite different domains (e.g. √x2 vs x, |x| vs x).→
- Induced domain droppedRecovers f(x) from f(g(x)) but omits the domain restriction from the inner substitution (e.g. |t|≥2 for x+1/x).→
- Log-argument positivity ignoredForgets the logarithm argument must be strictly positive.→
- Non-integer-power base not positiveAllows a zero or negative base under a non-integer/negative fractional exponent.→
- Radicand inequality direction wrongUses the wrong sign for the under-radical nonnegativity condition.→
- Second functional equation not formedFails to substitute 1-x (or the symmetric argument) to build the second equation needed to solve for f(x).→
- t=2x positivity forgottenSubstitutes t=2x to get a quadratic but forgets t must be positive when back-solving for the domain.→
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