domain of a function

Question

Find the domain of each function. (using interval notation)

1) f(x)= 1-e^x^2 / 1-e^1-x^2

2) f(x)= 7+x/ e^cos(x)

Answer 1

(1). In the case of fractions, the domain is restricted by the denominator. The function is undefined when the denominator is zero. In this case 1-e^1-x²=0 when e^1-x²=1, that is, when 1-x²=0=(1-x)(1+x).

At x=1 or x=-1 the function is undefined so the domain is {(-∞,-1), (-1,1), (1,∞)}.

(2) e^cos(x)=0 can never happen, so the domain is unrestricted: (-∞,∞).