For f(x) = (e^x + 1)/(e^x - 1), let n(d) be the number of integers not in its domain and n(r) be the number of integers not in its range. Find n(d) + n(r).

Question

Accepted Answer

4. Denominator e^x - 1 = 0 only at x = 0, so the only integer missing from the domain is 0, giving n(d) = 1. For the range, set y = (e^x+1)/(e^x-1) and solve: e^x = (y+1)/(y-1). Requiring e^x > 0 (and e^x not equal 1) forces y < -1 or y > 1, so the range is (-inf,-1) U (1, inf). The integers absent from the range are those in [-1, 1], namely -1, 0, 1, so n(r) = 3. Therefore n(d) + n(r) = 1 + 3 = 4.

For f(x) = (e^x + 1)/(e^x - 1), let n(d) be the number of integers not in its domain and n(r) be the number of integers not in its range. Find n(d) + n(r).

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