Problem 45 - Algebra 2

Question

Graph. Find the domain and the range of each function. \(y=-3 \sqrt[3]{x-4}-3\)

Step-by-Step Solution

Answer

The domain and range of the function \(y=-3 \sqrt[3]{x-4}-3\) are as follows: Domain: \(-\infty < x < \infty\), Range: \(-\infty < y < \infty\).

Step 1: Identify Domain

For a cubic function, the domain is all real numbers. So, regardless of the value within the cube root, the domain in this case will be all real numbers. Thus, Domain: \(-\infty < x < \infty\)

Step 2: Identify Range

Meanwhile, the range of the function is also all real numbers due to the nature of the cube root function. Both negative and positive y-values are possible. However, given that the function is being multiplied by -3 and then subtracted by 3, there will be a vertical stretch and shift downward, but it will not limit the scope of y-values the function can take. Thus, Range: \(-\infty < y < \infty\)

Step 3: Graphing the Function

The function, given its cubic root base, will have the basic shape of a cubic root function. It grows from left to right, and crosses the y-axis at y = -3 due to the subtraction operation. Likewise, due to the -3 multiplier, the function is reflected vertically (flipped over the x-axis). This graphical representation further confirms the domain and range determined in the previous steps.

Key Concepts

Domain of a Function

When discussing the domain of a function, we are trying to determine all the possible input values (x-values) that a function can accept. Some functions have restricted domains, meaning that there are certain values of x that don't work in the function. For instance, in functions involving square roots or fractions, these restrictions can come into play.
However, when dealing with cube root functions, like the one in our exercise, it's rather straightforward. The cube root function, regardless of its transformations, can accept any real number as an input.
This means the domain is all real numbers, expressed in mathematical terms as

• \( -\infty < x < \infty \)

This implies you can plug in any real number into the function, and it will give you a valid output without complications.

Range of a Function

The range of a function is somewhat the opposite of the domain. It refers to all the possible output values (y-values) that a function can produce. Understanding the range helps us know the extent of what the function can represent visually on a graph.
Cube root functions naturally have a range of all real numbers. This arises from the fact that you can input any real number into a cube root, and it will stretch infinitely in both the positive and negative y-direction.
Even though our specific exercise involves transformations - a vertical stretch by -3 and a downward shift by -3 - the essential nature of the cube root is not restricted. Therefore, the range remains all real numbers as well, represented as

• \(-\infty < y < \infty\)

These transformations affect the graph's shape and position but do not limit the potential y-values the function can achieve.

Cube Root Functions

Cube root functions form a distinctive type of function, centered around the cube root operation. The expression \( \sqrt[3]{x} \) can be recognized as its fundamental form. These functions have unique characteristics that differentiate them from other root functions, notably:

• They are continuous and defined for all real numbers.
• They produce an output for every real input.
• They have no breaks, gaps, or holes in their graphs.

For the exercise at hand, we looked at a more complex version of a cube root function: \(y=-3 \sqrt[3]{x-4}-3\). Such modifications introduce advanced behavior, like:

• The \'-3\' multiplier results in a vertical reflection and stretch, flipping the graph over the x-axis and making it steeper.
• The \'-4\' inside the cube root shifts the graph horizontally to the right by 4 units, indicating the function's new center.
• Finally, \'-3\' subtracted outside the root shifts the graph downward by 3 units.

Understanding these transformations can help when attempting to graph these functions, ensuring you accurately adjust for shifts and reflections described by the equation.