Giving your answer in interval notation, find the domain of the function sqrt(((x+4)(x−4))/((x−7)(x−5)))
√( ((x+4)(x−4)) / ((x−7)(x−5)) )
First, you need to exclude values which make the denominator zero: 5 and 7
Next, since the domain of √x is x>=0, you need
((x+4)(x−4)) / ((x−7)(x−5)) >= 0
This is true only for
x <= -4, 4<=x<5 and x>7
or
(-∞,-4]U[4,5)U(7,∞)
To find the domain of the function, we need to consider the values of x for which the function is defined. In other words, we need to identify any values of x that could make the function undefined by causing division by zero or evaluating a square root of a negative number.
In this case, the function involves division and square roots. Let's break it down step by step to identify any restrictions.
1. Division by zero: We need to check if there are any values of x that would make the denominator zero. In the given function, the denominator is (x - 7)(x - 5). To avoid division by zero, we need to ensure that x ≠ 7 and x ≠ 5. Therefore, the restrictions for division are x ≠ 7 and x ≠ 5.
2. Square root of a negative number: We need to check if there are any values of x that would result in taking the square root of a negative number. In this function, the square root is applied to ((x + 4)(x - 4)). For this expression to be defined, we need to ensure that it is greater than or equal to zero. In other words, we need to solve the inequality ((x + 4)(x - 4)) ≥ 0.
To solve this inequality, we can create a sign chart by considering the critical points: x = -4 and x = 4, which divide the number line into three intervals.
When x < -4, both factors (x + 4) and (x - 4) are negative, so their product is positive: ((x + 4)(x - 4)) > 0.
When -4 < x < 4, (x + 4) is positive while (x - 4) is negative, so their product is negative: ((x + 4)(x - 4)) < 0.
When x > 4, both factors (x + 4) and (x - 4) are positive, so their product is positive: ((x + 4)(x - 4)) > 0.
In summary:
- For ((x + 4)(x - 4)) > 0, we have two intervals: (-∞, -4) ∪ (4, +∞).
- For ((x + 4)(x - 4)) < 0, we have one interval: (-4, 4).
Combining the restrictions from division and square roots, the domain of the function is the intersection of these intervals, excluding -4, 4, 5, and 7. Therefore, the domain in interval notation is (-∞, -4) ∪ (-4, 4) ∪ (4, 5) ∪ (5, 7) ∪ (7, +∞).