what is the domain of the function f given f(x)= (sqrt(x ^ 2 - 4))/(x - 3) ?
The domain of a function is the set of all possible input values for which the function is defined. In this case, the function f(x) is defined as:
f(x) = (sqrt(x^2 - 4))/(x - 3)
To determine the domain, we need to consider any restrictions that may arise from the expression inside the square root and the denominator.
1. Expression inside the square root:
For the square root to be defined, the expression inside it must be greater than or equal to 0. So, we need to solve the inequality:
x^2 - 4 ≥ 0
(x - 2)(x + 2) ≥ 0
The critical points of this inequality are x = -2 and x = 2. We need to consider the sign of the expression in each of the intervals formed by these points:
1.1 For x < -2: We choose a test value of x, let's say x = -3. Plugging it into the expression (x - 2)(x + 2), we get (-3 - 2)(-3 + 2) = (-5)(-1) = 5, which is positive. So, the expression is positive in this interval.
1.2 For -2 < x < 2: We choose a test value of x, let's say x = 0. Plugging it into the expression (x - 2)(x + 2), we get (0 - 2)(0 + 2) = (-2)(2) = -4, which is negative. So, the expression is negative in this interval.
1.3 For x > 2: We choose a test value of x, let's say x = 3. Plugging it into the expression (x - 2)(x + 2), we get (3 - 2)(3 + 2) = (1)(5) = 5, which is positive. So, the expression is positive in this interval.
To satisfy the inequality, the expression inside the square root needs to be either positive or 0. Therefore, the valid intervals for x are: (-∞, -2] and [2, +∞).
2. Denominator:
The denominator of the function is (x - 3). For the function to be defined, the denominator must not be equal to 0. So, we exclude x = 3 from the domain.
Combining the results from both parts, the domain of the function f(x) = (sqrt(x^2 - 4))/(x - 3) is:
(-∞, -2] ∪ (2, 3) ∪ (3, +∞)
To determine the domain of the function f(x) = (sqrt(x^2 - 4))/(x - 3), we need to identify any values of x that would make the expression invalid.
Notice that the expression includes a square root. In order for the square root to be defined, the value inside the square root (x^2 - 4) must be greater than or equal to 0. Therefore, we have the inequality:
x^2 - 4 ≥ 0
To solve this inequality, we can factor it as follows:
(x - 2)(x + 2) ≥ 0
From this factorization, we can see that the inequality is satisfied when either:
1) (x - 2) ≥ 0 and (x + 2) ≥ 0, or
2) (x - 2) ≤ 0 and (x + 2) ≤ 0
For case 1), we have:
x - 2 ≥ 0 ⇒ x ≥ 2
x + 2 ≥ 0 ⇒ x ≥ -2
For case 2), we have:
x - 2 ≤ 0 ⇒ x ≤ 2
x + 2 ≤ 0 ⇒ x ≤ -2
Now, combining the solutions from both cases, we find that the values of x that satisfy the inequality are:
x ≤ -2 or x ≥ 2
However, we also need to consider the denominator (x - 3). Since division by zero is undefined, we cannot have x = 3.
Therefore, the domain of the function f(x) = (sqrt(x^2 - 4))/(x - 3) is:
x ≤ -2 or -2 < x < 3 or x > 3.