Digital SATMath: Advanced MathHard
A chemist is studying a reaction where the concentration of a reactant, C(t), in moles per liter, changes over time t, in seconds. The relationship is given by C(t) = (t^2 - 4t + 4) / (t^2 - 2t). For what value(s) of t is the concentration undefined?
- At = 0 and t = 2
- Bt = 2 only
- Ct = 0 only
- DThe concentration is defined for all t ≥ 0.
Show answer & explanationAnswer & explanation
Correct answer: A. t = 0 and t = 2
A rational function is undefined when its denominator is zero. Set the denominator t^2 - 2t = 0. Factor out t: t(t - 2) = 0. This yields t = 0 or t = 2. Both values make the function undefined.
Why the other options are wrong
- B. This only identifies one of the values that makes the denominator zero.
- C. This only identifies one of the values that makes the denominator zero.
- D. The function is undefined at specific points where the denominator is zero, so it's not defined for all t ≥ 0.
Domain of Rational Functions
The domain of a rational function consists of all real numbers except those values for which the denominator is equal to zero. These values lead to vertical asymptotes or holes.
- Set the denominator equal to zero to find excluded values.
- Factor the denominator to find all roots.
- The numerator does not affect the domain, only the zeros of the function.
Memory trick: Denominator Zero, Domain No Go!