91Ó°ÊÓ

A square root function is any function that includes a square root symbol. For example, in our exercise, the function is expressed as \( y = \sqrt{3 x - 1} + 4 \).
Square root functions have a unique property: the expression under the square root (called the radicand) must be greater than or equal to zero. This is because the square root of a negative number is not a real number.
For example, in our function, the radicand is \( 3x - 1 \). To make sure our function is valid, we must ensure \( 3x - 1 \geq 0 \). From this principle, you can determine the domain of the function, which tells you the values of \( x \) for which the function is defined.

Solving inequalities is a fundamental skill in algebra and is key to finding the domain of functions involving square roots.
In our specific function \( y = \sqrt{3x - 1} + 4 \), we set the radicand \( 3x - 1 \) to be greater than or equal to zero: \( 3x - 1 \geq 0 \).
Here's the process:

• First, we add 1 to both sides: \( 3x \geq 1 \).
• Next, we divide both sides by 3: \( x \geq \frac{1}{3} \).

The solution to the inequality, \( x \geq \frac{1}{3} \), provides the domain of the function. This means the function is only defined for values of \( x \) that are \( \frac{1}{3} \) or greater.

Determining the range of a function involves finding the possible values that the function can take.
For the function \( y = \sqrt{3x - 1} + 4 \), it is important to understand that the square root function itself only produces non-negative results. This is because the square root of zero or any positive number is always non-negative.
The minimum value of \( \sqrt{3x - 1} \) is 0, which happens when \( 3x - 1 = 0 \), or equivalently \( x = \frac{1}{3} \). Adding 4 to this minimum value of 0 results in 4.
Since the square root function can continue to increase indefinitely as \( x \) increases, the highest value of \( y \) is infinite. Hence, the range of our function is \( [4, \infty) \). This means \( y \) can take any value starting from 4 and extending to infinity.