Question

Find the domain and the range of the function. $$f(x)=\sqrt{x-8}$$

Short answer

The domain of the function is \(x \ge 8\) and the range of the function is \(y \ge 0\).

Step-by-step solution

Find the domain of the function

To find the domain, set the value inside the square root \(x-8\) greater than or equal to zero and solve for x: \(x-8 \ge 0\) Adding 8 to both sides of the equation gives: \(x \ge 8\) Hence, the domain of the function is \(x \ge 8\).

Find the range of the function

The range of the function is the set of all possible output values. Since this is a square root function, the output will be non-negative (i.e., greater than or equal to zero). Hence, the range of the function is \(y \ge 0\).

Key concepts

Square Root Function

When studying functions in algebra, the square root function is a fundamental concept that takes the form of \( f(x) = \sqrt{x} \). This function represents the principal square root of \( x \). The most important aspect to consider for such a function is its domain—what values of \( x \) we're allowed to plug into the function without causing mathematical issues.

For the given function \( f(x) = \sqrt{x-8} \), we're dealing with the square root of an algebraic expression \( x-8 \). To not end up taking the square root of a negative number (which would give us complex numbers, not covered in basic algebra), we set the inside of the square root to be greater than or equal to zero: \( x-8 \ge 0 \). Solving this inequality gives us \( x \ge 8 \), defining the domain where the function can operate safely with real numbers. Any number below 8 is not included in the domain because it would create a negative number under the square root, hence it's impossible for the function.

Inequalities

Inequalities are mathematical expressions involving the symbols \( > \), \( < \), \( \ge \), or \( \le \), which denote 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', respectively. They are used to describe the relative size of two values.

Understanding how to solve inequalities is crucial when finding the domain and range of functions. As demonstrated in the exercise, we solved the inequality \( x-8 \ge 0 \) to determine the domain of the square root function. The domain here is all values of \( x \) that satisfy the condition \( x \ge 8 \). When it comes to square root functions or any functions involving square roots, ensuring that the value under the square root is non-negative is a foundational use of inequalities. This process involves similar steps to solving equalities, but it is essential to remember that multiplying or dividing by a negative number will reverse the inequality's direction.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \) or \( y \)), and arithmetic operations (such as \( + \), \( - \), \( * \), and \( / \)) without an equality sign. They're the building blocks of algebra and help us describe patterns, construct functions, and model real-world situations.

In our function \( f(x) = \sqrt{x-8} \), the expression \( x-8 \) is an algebraic expression. It consists of the variable \( x \) and a constant number \( -8 \). Manipulating this algebraic expression by applying operations such as adding, subtracting, and taking square roots is central to solving and understanding the behavior of functions in algebra. By finding the domain of \( f(x) \), we altered the algebraic expression to ensure that it would produce a real number after taking its square root, exemplifying how algebraic expressions are adapted and constrained to satisfy different mathematical rules and conditions.