: Absolute Value Divided by a Linear Factor

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

: Absolute Value Divided by a Linear Factor (a) Define q:R∖{2}→Rq:\mathbb{R}\setminus\{2\}\to\mathbb{R} by q(x)=∣x2−4∣x−2\displaystyle q(x)=\dfrac{|x^{2}-4|}{x-2}. Answer the following: (b) (i) Express q(x)q(x) as a piecewise rule without the absolute-value sign, indicating the interval of xx to which each expression applies. (c) (ii) For every interval named in part (i), classify the resulting expression as polynomial, constant, rational or modulus. (d) (iii) Simplify each expression wherever possible and state all real numbers (other than 22) at which qq is not defined. (e) (iv) Determine whether it is possible to define a value q(2)q(2) so that the extended rule becomes continuous at x=2x=2. If it is possible, state that value and justify your answer; if not, give a reason.

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