how do you find the domain of the function::
14/(x^2 - 361) <-- this is all under a square root..
i have attempted this so many times and have not found the correct answer yet. how do i solve the domain when it is a square root function with a denominator???? please help!
thanks
Two conditions are the problem here:
1. We cannot divide by zero
2. We cannot take the square root of a negative.
so 14/√(x^2 - 361)
= 14/√[(x+19)(x-19)]
so clearly x cannot be ± 19
secondly, wouldn't the number under the root sign be negative for all values between 1- and +19
so the domain is
x > 19 OR x < -19 , x is a real number.
To find the domain of the given square root function, you need to consider the restrictions that could result in undefined values. In this case, the denominator cannot be equal to zero since dividing by zero is undefined.
The square root function itself allows only non-negative values, so the expression inside the square root, x^2 - 361, must be greater than or equal to zero.
To find the domain, follow these steps:
Step 1: Set the denominator equal to zero to find any potential restrictions on x. In this case, x^2 - 361 = 0. Solving this equation gives us x = ±19.
Step 2: The denominator cannot be zero, so we must exclude these values from the domain. Therefore, the domain excludes x = -19 and x = 19.
Step 3: Consider the expression inside the square root. Since it is x^2 - 361, the expression must be greater than or equal to zero.
Step 4: To find the values of x that make the expression x^2 - 361 greater than or equal to zero, you can solve the inequality x^2 - 361 ≥ 0.
Step 5: Factor the quadratic expression x^2 - 361 = 0. You get (x - 19)(x + 19) ≥ 0.
Step 6: Determine the sign of the quadratic expression between the critical points x = -19 and x = 19 by testing a value in each region. For example, test x = 0: (0 - 19)(0 + 19) ≥ 0 → (-19)(19) ≥ 0 → -361 ≥ 0. Since this is not true, the expression is negative between -19 and 19.
Step 7: Since a square root only allows non-negative values, the expression x^2 - 361 must be greater than or equal to zero. This means that x^2 - 361 ≥ 0 when x ≤ -19 or x ≥ 19.
Step 8: Combine the restrictions found in Step 2 with the range found in Step 7. Therefore, the domain of the function is x ∈ (-∞, -19) ∪ (-19, 19) ∪ (19, +∞).
In summary, the domain of the given square root function is all real numbers except x = -19 and x = 19.