91Ó°ÊÓ

Understanding "domain" and "range" is key when exploring functions. The domain refers to all possible input values (often denoted as x-values) for which the function is defined. For the given function, \(y = 4 \sqrt[3]{x-2} + 1\), the domain is all real numbers, expressed as \(D=(-\infty, \infty)\) or \(D=\mathbb{R}\).
This is because the cubic root, unlike a square root, exists for both negative and positive numbers. Hence, every value of x works here.
On the other hand, the "range" is the set of possible output values (y-values). For our function, as x takes on any real number, y can also be any real number. This is why the range is \(R=(-\infty, \infty)\) or \(R=\mathbb{R}\).
In simpler terms, the function stretches infinitely in both directions along the y-axis. Knowing the domain and range helps in knowing the entire behavior of the function.

When graphing a function like \(y = 4 \sqrt[3]{x-2} + 1\), you're visually representing how y changes with respect to x. Start by plotting some key points. Substitute various x-values into the function to find corresponding y-values.
An effective way is to choose easy numbers for x, like 0, 1, and 2, and calculate y. Then plot these pairs on a coordinate plane.

• For \(x = 2\), \(y = 4(\sqrt[3]{2-2}) + 1 = 1\).
• For \(x = 3\), \(y = 4(\sqrt[3]{3-2}) + 1 = 5\).
• For \(x = 1\), \(y = 4(\sqrt[3]{1-2}) + 1 = -1\).

After plotting these points, sketch the curve smoothly through them. You'll notice the graph is continuous and extends infinitely in both the upward and downward directions, in line with the domain and range. Graphing confirms how the function behaves and whether interpretations are correct.

Cubic root functions, like \(y = 4 \sqrt[3]{x-2} + 1\), feature unique characteristics from other root functions. They calculate the cube root of a number, denoted by \(\sqrt[3]{x}\). Unlike square root functions, cubic roots can handle negative and positive values of x without any issues.
This property means cubic root functions are defined over all real numbers. In our function, x can be as small or large as desired, and y will still have a valid solution.
A cubic root function has an S-shaped curve on its graph. Its steep increase or decrease represents rapid changes in y when plotted against x. Changes such as adding 4 and shifting by 1, as seen in \(y = 4 \sqrt[3]{x-2} + 1\), affect the stretch and vertical shift of the graph. This makes it rise more sharply and start from a y-value of 1 instead of zero. Understanding these properties helps you grasp the dynamics and transformation of cubic root functions on a graph.