Part 1Find the domain of the function f(x)equals=StartFraction x squared minus 4 Over x squared minus 9 EndFractionx2−4x2−9. Part 1The domain is [input]enter your response here .(Type your answer in interval notation.)

Question

Part 1Find the domain of the function ​f(x)equals=StartFraction x squared minus 4 Over x squared minus 9 EndFractionx2−4x2−9. Part 1The domain is [input]enter your response here .​(Type your answer in interval​ notation.)

BlackTom AI · Accepted Answer

We start by identifying the domain of the rational function f(x) = (x^2 - 4) / (x^2 - 9).

Step 1: Factor numerator and denominator. The numerator factors as x^2 - 4 = (x - 2)(x + 2). The denominator factors as x^2 - 9 = (x - 3)(x + 3).

Step 2: Determine where the function is defined. A rational function is undefined where the denominator equals zero. So we set (x - 3)(x + 3) = 0, giving x = 3 and x = -3 as the points to exclude from the domain.

Step 3: Check for potential cancellations. There are no common factors between the numerator (x - 2)(x + 2) and the denominator (x - 3)(x + 3). Therefore, we cannot cancel any factors to extend the domain.

Step 4: Conclude the domain. All real numbers except x = -3 and x = 3 are allowed. In interval notation, the domain is (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).

If you plug in any value not equal to ±3, the denominator is nonzero, so the expression yields a real number; at x = ±3 the function is undefined due to division by zero.

Part 1Find the domain of the function f(x)equals=StartFraction x squared minus 4 Over x squared minus 9 EndFractionx2−4x2−9. Part 1The domain is [input]enter your response here .(Type your answer in interval notation.)Multiple fill-in-the-blank

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Part 1Find the domain of the function ​f(x)equals=StartFraction x squared minus 4 Over x squared minus 9 EndFractionx2−4x2−9. Part 1The domain is [input]enter your response here .​(Type your answer in interval​ notation.)Multiple fill-in-the-blank