Problem 51 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=\sqrt{x-5} $$

Step-by-Step Solution

Answer

The domain of the function is \([5, \infty)\).

Step 1: Understand the Problem

To find the domain of a function, we need to determine the set of all possible input values (x-values) that make the function defined. For the function \( f(x) = \sqrt{x-5} \), which includes a square root, we need to ensure that the expression inside the square root is non-negative.

Step 2: Set Up the Inequality

For the square root function \( f(x) = \sqrt{x-5} \) to be defined, the expression under the square root \( x - 5 \) must be greater than or equal to zero. Therefore, set up the inequality: \[ x - 5 \geq 0 \].

Step 3: Solve the Inequality

Solve the inequality \( x - 5 \geq 0 \) to determine the allowable values for \( x \). Add 5 to both sides to get: \[ x \geq 5 \]. This inequality tells us that \( x \) must be greater than or equal to 5 for the function to be defined.

Step 4: Express the Domain

Express the domain of the function in interval notation based on the inequality \( x \geq 5 \). The domain of \( f(x) = \sqrt{x-5} \) is all values of \( x \) starting from 5 and extending to infinity. Therefore, the domain is \([5, \, \infty)\).

Key Concepts

Square root functions

Square root functions are a special type of function in algebra. They involve the square root of an expression and are typically represented as \( f(x) = \sqrt{x} \). One key aspect of square root functions is that they are only defined for non-negative numbers. This means that whatever is inside the square root must be either zero or positive because you cannot take the square root of a negative number using real numbers.

For example, in the function \( f(x) = \sqrt{x-5} \), the expression \( x-5 \) must be greater than or equal to zero. This condition ensures that the square root is defined and that the function can produce real number outputs. When solving problems with square root functions, it's crucial to set up an inequality to determine where the expression inside the square root is non-negative, just like in the exercise above.

Inequalities in algebra

Inequalities are mathematical expressions involving symbols like \( <, \leq, >, \) and \( \geq \). In algebra, these symbols are used to represent the relationship between two quantities. Inequalities tell us about the range of values that could make an expression true. They have a critical role in determining the domain of functions, especially when square roots are involved.

For the square root function \( f(x) = \sqrt{x-5} \), we used the inequality \( x - 5 \geq 0 \) to determine the values of \( x \) that make the function defined. This technique involves a few simple steps:

• Identify the expression inside the square root.
• Set up an inequality that ensures this expression is non-negative, such as \( x - 5 \geq 0 \).
• Solve the inequality to find the solution. For instance, by adding 5 to both sides, we get \( x \geq 5 \).

This result means that \( x \) has to be at least 5 for the square root function to have real number outputs.

Interval notation

Interval notation is a concise way to express a range of numbers, often used to describe domains of functions and solutions to inequalities. It uses brackets and parentheses to show whether endpoints are included or not.

In interval notation:

• Brackets, like [ or ], indicate that an endpoint is included in the interval. For example, \([5, \infty)\) means 5 is included.
• Parentheses, like ( or ), indicate that an endpoint is not included. \( (5, \infty) \) would mean that 5 is not included, though infinity will always have a parentheses.

When we solved the inequality \( x \geq 5 \) for the function \( f(x) = \sqrt{x-5} \), we translated this into interval notation as \([5, \infty)\). This notation tells us that x can take any value starting from 5 and going upwards to infinity, but it cannot be less than 5. Understanding interval notation is crucial as it simplifies the representation of domains and ranges in functions.