What is the domain of the function f(x)=sqrt(x^4-16)? In interval notation. Please show me how to work this problem out. I cannot find any examples in my book. Thanks!
You cannot take the square root of a negative. So f(x) is defined when the contents of the square root are greater than or equal to 0.
x^4 - 16 >= 0
x >= ? and x <= ?
Interval notation expresses an included endpoint (greater than or equal) with a "[]" and a not included endpoint (greater than) with a "()"
For example, if the domain of x is -2 < x <= 3, then interval notation would be (-2,3].
To find the domain of a function, you need to determine the set of all valid inputs (values of x) that the function can accept without resulting in any undefined outputs. In this case, the function is f(x) = √(x^4 - 16).
To find the domain of f(x), we have to consider two factors: the square root function (√) and the expression inside it (x^4 - 16). Let's break it down step by step:
1. We start by considering the expression inside the square root (√).
x^4 - 16 ≥ 0
The reason for this is that the square root of a negative number is undefined in the real number system, so we need to ensure that x^4 - 16 is non-negative.
2. Now we solve the inequality x^4 - 16 ≥ 0 to find the valid values for x:
x^4 - 16 ≥ 0
One way to solve this inequality is to factor it:
(x^2 - 4)(x^2 + 4) ≥ 0
Next, we find the critical points by setting each factor equal to zero:
x^2 - 4 = 0 and x^2 + 4 = 0
Solving these equations, we get two pairs of critical points:
x = -2, x = 2, x = -2i, x = 2i
These are the points where the inequality changes sign.
3. To determine the sign of the inequality for each interval, we can use a sign chart:
-∞ -2 2 +∞
(x^2 - 4)(x^2 + 4): + - + +
We check the sign of the inequality within each interval to find the solution.
For the interval (-∞, -2), both factors are positive (+), yielding a positive product (+).
For the interval (-2, 2), the first factor (x^2 - 4) is negative (-), while the second factor (x^2 + 4) is positive (+), resulting in a negative product (-).
For the interval (2, +∞), both factors are positive (+), giving a positive product (+).
4. Finally, we state the solution in interval notation:
The domain of the function f(x) = √(x^4 - 16) is (-∞, -2] ∪ [2, +∞).
Therefore, the domain of the function f(x) = √(x^4 - 16) is (-∞, -2] ∪ [2, +∞).