Question

31\. What is the domain of the function \(f(x)=\sqrt{x^{2}-16}\) ?

Short answer

The domain of the function is \( (-\infty, -4] \cup [4, \infty) \).

Step-by-step solution

Identify the function type

The function is a square root function: \( f(x) = \sqrt{x^2 - 16} \)

Set the radicand greater than or equal to zero

For the square root function \( \sqrt{A} \), the expression inside the square root \( A \) must be greater than or equal to zero to obtain real numbers. So, set \( x^2 - 16 \geq 0 \).

Solve the inequality \(x^2 - 16 \geq 0\)

Rewrite the inequality as \( x^2 \geq 16 \). This can be solved as \( x \geq 4 \) or \( x \leq -4 \).

Express the solution in interval notation

The solution \( x \geq 4 \) or \( x \leq -4 \) can be expressed in interval notation as \( (-\infty, -4] \cup [4, \infty) \).

Key concepts

square root function

Square root functions are types of functions that involve a square root operation. They are typically written as \( f(x) = \sqrt{A} \), where \( A \) is an expression involving the variable \( x \). What makes square root functions special is that the expression inside the square root, also known as the radicand, must be non-negative (greater than or equal to zero). This is because you cannot take the square root of a negative number and get a real number. Therefore, to determine the domain of a square root function, you need to find all the values of \( x \) that make the radicand non-negative.
In our example, the function is \( f(x) = \sqrt{x^2 - 16} \). To find its domain, we need to ensure the expression inside the square root, \( x^2 - 16 \), is greater than or equal to zero. This makes the function's output a real number.

inequality

An inequality is a mathematical statement that there is a relationship of inequality (greater than, less than, or equal to) between two expressions. For example, \( x^2 - 16 \geq 0 \) is an inequality. When you encounter an inequality in the context of square root functions, solving it helps you determine the range of values that the radicand can take.
To solve the inequality \( x^2 - 16 \geq 0 \), we first rearrange it as \( x^2 \geq 16 \). This tells us we are interested in all \( x \)-values where \( x^2 \) is at least 16. Next, we find the values of \( x \) that satisfy this condition. From basic algebra, we know that \( x^2 = 16 \) when \( x = 4 \) or \( x = -4 \). Therefore, values of \( x \) that make \( x^2 \geq 16 \) are either greater than or equal to 4 or less than or equal to -4.

interval notation

Interval notation is a way of writing the set of all numbers between two endpoints. It includes the use of square brackets \(( [ ] )\) and parentheses \(( ( ) )\) to denote whether endpoints are included or excluded from the interval.
For the solution to \( x^2 \geq 16 \), we determined \(x \geq 4 \) or \(x \leq -4 \). To express this in interval notation, you write the intervals where the condition is met. It translates to \((-\infty, -4] \cup [4, \infty)\). Here, \( ( -\infty, -4] \) indicates all numbers less than or equal to -4, and \([4, \infty) \) represents all numbers greater than or equal to 4. The union symbol \( \cup \) indicates the combined set of these intervals.
Thus, using interval notation, you can compactly represent the domain of the function \( f(x) = \sqrt{x^2 - 16} \) as all \( x \)-values that satisfy the inequality.