Find the Domain (1-e^(x^2))/(1-e^(1-x^2))

Problem

(1−e(x2))/(1−e(1−x2))

Solution

1. Identify the restriction for a rational function, which requires the denominator to be non-zero.

1−e(1−x2)≠0

2. Isolate the exponential term by moving it to the other side of the inequality.

e(1−x2)≠1

3. Apply the natural logarithm to both sides to solve for the exponent, noting that ln(1)=0

1−x2≠ln(1)

1−x2≠0

4. Solve the resulting quadratic equation for x

x2≠1

x≠±1

5. Express the domain in interval notation, excluding the values where the denominator is zero.

(−∞,−1)∪(−1,1)∪(1,∞)

Final Answer

Domain: *{[x∈ℝ,x≠±1]}

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