Question

Find the domain of the function. $$y=5-\sqrt{x}$$

Short answer

The domain of the function \(y=5-\sqrt{x}\) is \([0, +\infty)\). In other words, \(x\) can be any real number greater than or equal to 0.

Step-by-step solution

Identify function type

The function given is a square root function. In square root functions, the value under the square root sign, also known as the radicand, should be greater than or equal to 0 to avoid complex roots.

Set up Inequality

Given that the value under the square root sign should be greater than or equal to 0, set up an inequality where \(x\) is greater than or equal to 0. The inequality is \(x \geq 0\).

Solve for the domain

Solving the inequality \(x \geq 0\) gives the solution as \(x\) is any real number that is greater than or equal to 0. This solution set then forms the domain of the given function.

Key concepts

Square Root Function

A square root function includes the operation of taking the square root of a number. It is characterized by the presence of a radical sign \( \sqrt{} \) with a quantity underneath it, known as the radicand. In the function \( y = 5 - \sqrt{x} \), \( x \) is the radicand.

For the square root of \( x \) to be a real number, \( x \) cannot be negative because the square root of a negative number is not defined in the set of real numbers. Therefore, the domain of a square root function includes all the values of \( x \) that makes the radicand non-negative (\( x \geq 0 \)). In simple terms, you can only take the square root of zero or positive numbers if you wish to stay within the realm of real numbers.

Understanding square root functions is crucial for determining their domains since it instills the concept that these functions can only handle non-negative inputs to yield real number outputs.

Inequality

An inequality is a mathematical statement that describes the relation of one quantity with another, indicating that they are not equal but rather 'less than' or 'greater than' each other.

In the case of a square root function, establishing the domain involves setting up an inequality where the radicand is greater than or equal to zero. This ensures the output stays within the real numbers. The inequality \( x \geq 0 \) is a simple one, encompassing all numbers from zero to infinity, including zero itself. This step is about ensuring that no square root of a negative number is taken, which would lead to non-real results.

Grasping the concept of inequalities allows students to solve various domain-related problems by understanding the number sets that satisfy the conditions required by different functions.

Real Numbers

The set of real numbers is a fundamental concept in mathematics, encompassing all the numbers we typically use, including rational numbers (like \( \frac{1}{2} \), \( 3 \)) and irrational numbers (like \( \sqrt{2} \), \( \pi \)). In simple terms, the real numbers are all the points on an infinitely long number line.

In relation to the domain of a function, real numbers are significant because functions typically map inputs from the set of real numbers to outputs also in the set of real numbers. For the function \( y = 5 - \sqrt{x} \), our domain consists of all the real numbers greater than or equal to zero because this guarantees the output will also be a real number. Thus, understanding the set of real numbers helps in identifying valid inputs for functions and ensuring that their outputs remain within the realm of practical and measurable values.