Question

Find the domain of the function. $$y=\sqrt{x}-3$$

Short answer

The domain of the function \(y=\sqrt{x}-3\) is all real numbers that are greater than or equal to 0. This can also be written as \([0, \infty)\).

Step-by-step solution

Recall the rule of the Square Root Function

Remember that for a square root function, the value inside the square root (radicand) must be greater than or equal to 0. This is because the square root of a negative number isn't valid in the realm of real numbers.

Apply the Square Root Rule to the Function

In order to apply the square root rule, set the radicand (x) to be greater than or equal to 0. This looks like: \(x \ge 0\).

Identify Domain from Inequality

The inequality \(x \ge 0\) represents all possible values of \(x\) that will make the function valid. Thus, the domain of the function \(y=\sqrt{x}-3\) are all real numbers that are greater than or equal to 0.

Key concepts

Square Root Function

The square root function is a fundamental concept in mathematics and is represented by \( y = \sqrt{x} \). At its core, the square root function asks the question, 'what number, when multiplied by itself, will give me \( x \)?'. It's essential to understand that square roots have a very specific requirement: they can only be taken of non-negative numbers in the realm of real numbers. So, for \( y = \sqrt{x} \) to be a valid operation, \( x \) must be greater than or equal to zero.

This is because, in real numbers, the square root of a negative number doesn't exist; such an operation enters the realm of complex numbers. This limitation on the radicand heavily influences the domain of square root functions. The graph of the square root function is a curve starting from the origin (when \( x = 0 \) and \( y = 0 \) by definition) and increasing gradually, never decreasing or turning back on itself.

Radicand

The term 'radicand' refers to the number or expression inside a radical symbol, which in the square root function's case is the symbol \( \sqrt{\phantom{x}} \). For our function \( y = \sqrt{x} - 3 \) the radicand is simply \( x \) itself. The value of this radicand dictates whether the square root function can produce a real number result.

In the context of the provided exercise, we focus on ensuring that the radicand does not lead us into the realm of complex numbers. Thus, when we investigate the domain of a square root function, our central concern is identifying the set of all possible real numbers for the radicand that will make the function 'work'. We do this by setting an inequality that the radicand must satisfy, which often results in a range of numbers that form the function's domain.

Inequality

Inequalities are mathematical expressions that describe the relative size or order of two values. They are widely used to define intervals or ranges of numbers. In the context of square root functions, inequalities help us establish the domain by setting conditions on the radicand.

For our exercise example \( y = \sqrt{x} - 3 \) the inequality \( x \ge 0 \) is utilized to express that \( x \) must be greater than or equal to zero. This inequality not only tells us what numbers we cannot use (anything less than zero), but also implicitly defines the domain of the function - all real numbers zero or greater. The inequality symbol \( \ge \) is crucial as it includes zero itself, reflecting the fact that zero has a valid square root: zero.