Find the Domain ( square root of 4-x)/((x+1)(x^2+1))

Problem

√(,4−x)/((x+1)*(x2+1))

Solution

1. Identify the restriction for the square root in the numerator. The expression inside a square root must be greater than or equal to zero.

4−x≥0

2. Solve the inequality for the numerator restriction.

−x≥−4

x≤4

3. Identify the restriction for the denominator. The denominator cannot be equal to zero.

(x+1)*(x2+1)≠0

4. Solve for the values that make the denominator zero. Set each factor to zero.

x+1=0⇒x=−1

x2+1=0⇒x2=−1

5. Analyze the results of the denominator equations. The equation x2=−1 has no real solutions. Therefore, the only real value that makes the denominator zero is x=−1

x≠−1

6. Combine all restrictions. The domain consists of all real numbers x such that x≤4 and x≠−1

Final Answer

Domain:(−∞,−1)∪(−1,4]

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