Question

Find the domain and the range for each function.$$y=\frac{3}{\sqrt{x-2}}$$

Short answer

The domain of the function \(y=\frac{3}{\sqrt{x-2}}\) is \(x > 2\), and the range is \(y > 0\).

Step-by-step solution

Determine the Domain

The domain of a function is the set of all possible input values (x-values) that allow the function to produce a real number output. For the function \(y=\frac{3}{\sqrt{x-2}}\), the denominator \(\sqrt{x-2}\) cannot be zero or negative, since you cannot take the square root of a negative number. Therefore, the expression inside the square root, \(x-2\), must be greater than zero: \(x-2 > 0\). Solving for x, we find the domain: \(x > 2\).

Determine the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. In our function, since \(x\) is always greater than 2, \(\sqrt{x-2}\) will always be greater than 0. Therefore, the denominator of our function is always positive, implying the function values are always positive as well. In other words, the range is the set of all positive real numbers: \(y > 0\).

Key concepts

Function Real Number Output

When we refer to a function's real number output, we're talking about the results a function yields when a real number input is applied. In mathematics, functions are often defined as a relation between a set of inputs and a set of permissible outputs, with each input being related to exactly one output. For the function in the exercise, \(y=\frac{3}{\sqrt{x-2}}\), the output values are the real numbers you get after applying the square root operation and division to the input values from the function’s domain.

In this context, the key is to ensure that operations performed within the function, such as division by zero or taking the square root of a negative number, do not result in undefined or non-real numbers. This is crucial as non-real outputs (like complex numbers with an imaginary unit) are not part of the function's range when we are only considering real number outputs. Understanding that a function needs to produce a real number output helps us to define its domain and range correctly.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 (\(3^2 = 9\)). A major point to note about square roots in the realm of real numbers is that you can only take the square root of a non-negative number. This restriction is because squaring any real number, whether it is positive or negative, always results in a non-negative number.

In the given exercise \(y=\frac{3}{\sqrt{x-2}}\), the square root is applied to the expression \(x-2\). Since the square root of a negative number is not a real number, the values of \(x\) must be such that \(x-2\) is not negative, leading to the inequality \(x > 2\), which defines the function’s domain. The concept of a square root is crucial because it impacts the domain directly by excluding any \(x\)-values that would make the expression inside the root negative.

Inequalities

In mathematics, inequalities express the relation between two values that are not equal. They are written using inequality symbols such as \(>\), \(<\), \(\geq\), and \(\leq\). Solving inequalities is an essential skill because it helps us describe the domain and range of functions.

In the function \(y=\frac{3}{\sqrt{x-2}}\), we use an inequality to determine the domain— \(x-2 > 0\) means that \(x\) must be greater than 2. This is an example of how inequalities can define the set of numbers for which the function is valid. Similarly, understanding inequalities allows us to express the range: since the denominator \(\sqrt{x-2}\) is always positive for \(x > 2\), the output \(y\) is a positive real number. Therefore, we express the range using the inequality \(y > 0\). Learning to solve inequalities is vital in depicting the behavior of functions and setting the stage for finding their domains and ranges.