Question

Find the domain of each function. \(h(x)=\sqrt{3 x-12}\)

Short answer

Domain: \([4, \infty)\)

Step-by-step solution

- Identify the function type

The given function is a square root function: \[ h(x) = \sqrt{3x - 12} \]Square root functions are defined for non-negative values of the expression inside the square root.

- Set the radicand greater than or equal to zero

To find the domain, set the expression inside the square root (the radicand) to be greater than or equal to zero:\[ 3x - 12 \geq 0 \]

- Solve the inequality

Solve the inequality for \(x\):Add 12 to both sides:\[ 3x \geq 12 \]Divide both sides by 3:\[ x \geq 4 \]

- Write the domain

The domain of the function is the set of all \(x\) values for which the inequality holds true:\[ x \geq 4 \]In interval notation, this is \([4, \infty)\).

Key concepts

Square Root Function

To understand a square root function, let's break it down. A function like \( h(x) = \sqrt{3x - 12} \) is called a square root function because it includes the square root symbol. This means we only consider the non-negative 'square root' of the expression inside. For instance, \(\sqrt{9}\) gives 3, not -3, even though both numbers square to 9. The square root function always outputs positive values or zero.

The key idea is that what's inside the square root (called the 'radicand') must be zero or positive. In our example, \( 3x - 12 \) must be greater than or equal to zero, because square roots of negative numbers aren't allowed in real numbers. This is one of the most important aspects to remember about square root functions.

Inequalities

Inequalities are like equations but with symbols that show a range of values rather than a specific number. They use symbols like < (less than), > (greater than), \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to).

For the function \( h(x) = \sqrt{3x - 12} \), we need to find when \( 3x - 12 \) is greater than or equal to zero. This kind of inequality is written as \[ 3x - 12 \geq 0 \]. Solving it involves simple steps:

• Add 12 to both sides: \[ 3x \geq 12 \]
• Divide both sides by 3: \[ x \geq 4 \]

Now, we know \( x \) must be 4 or larger. This inequality has determined the range of valid \( x \) values for our function.

Interval Notation

Interval notation is used to describe a set of numbers between two endpoints. It's a concise way to show the realm of possible values. For example, \( [4, \infty) \) means all numbers from 4 to infinity, including 4 but not infinity because infinity is not a number, it's an idea of endlessly large values.

In our function \( h(x) = \sqrt{3x - 12} \), we found that \( x \geq 4 \). Therefore, the domain can be written in interval notation as \[ [4, \infty) \]. Here’s how to read it:

• \