Problem 47 - Algebra 2

Question

Graph. Find the domain and the range of each function. \(y=-\sqrt[3]{8 x}+5\)

Step-by-Step Solution

Answer

The domain of the function \(y=-\sqrt[3]{8x}+5\) is all real numbers (-∞,+∞), and the range of the function is also all real numbers (-∞,+∞).

Step 1: Determine the Domain

The domain of a function is the set of all possible input values (typically x-values) which will produce a valid output. Since there are no restrictions like division by zero or square root of negative numbers in this function, thus the domain of this function is all real numbers. In interval notation, this can be written as (-∞,+∞).

Step 2: Determine the Range

The range of a function is the set of possible output values (typically y-values). The original function \(\sqrt[3]{x}\) without transformations has a range of all real numbers. The given function is \(-\sqrt[3]{8x}+5\) which is the original cubic root function multiplied by -8 and then shifted up by 5. The multiplying by -8 reflects the function across y-axis and doesn't change its range which is still all real numbers. The upward shift by 5 also doesn't change the nature of range. Hence, the range is also all real numbers, or in interval notation, (-∞,+∞).

Key Concepts

Cube Root Function

The cube root function is a type of mathematical function that is closely related to the concept of cubing a number. In mathematical terms, the cube root of a number \( x \) is written as \( \sqrt[3]{x} \), and it represents the number which when cubed gives \( x \). For example, \( \sqrt[3]{27} = 3 \) because when you multiply 3 by itself three times (3 \( \times \) 3 \( \times \) 3), you get 27.

The cube root function itself looks different from the more common square root function. Unlike the square root which only deals with positive numbers and zero, the cube root is defined for all real numbers, including negative ones. This means that you can take the cube root of a negative number. For instance, \( \sqrt[3]{-8} = -2 \) because \(-2 \times -2 \times -2 = -8\).

This is why the domain and range of the cube root function \( y = \sqrt[3]{x} \) are both all real numbers; you can plug any real number into the function and produce any real number as output.

Graphing Functions

Graphing a function is a way to visually represent the values of a function. For the cube root function, the graph typically looks like an elongated 'S' shape that passes through the origin (0,0).

When we transform the function such as in \( y = -\sqrt[3]{8x} + 5 \), we change its appearance on a graph:

• Reflection: Multiplying the cube root by -8 reflects the original graph over the x-axis and stretches it vertically, since each value of \( y \) is now multiplied by -8.
• Vertical Shift: Adding 5 shifts the entire graph upward by five units. This means that every part of the graph is higher than it was in the original function \( y = \sqrt[3]{x} \).

Even with these transformations, the basic characteristics of the function's graph—such as it moving from the bottom left to the top right in an S-shaped curve—remain consistent. This means the overall behavior of the graph is similar to the original cube root function but with these specific transformations applied.

Real Numbers

Real numbers are a fundamental part of mathematics, encompassing both rational and irrational numbers. They include all the numbers on the number line, so they can be positive, negative, or zero.

Within the context of functions, real numbers play a crucial role because they usually define the domain and range. For instance, the function \( y = -\sqrt[3]{8x} + 5 \) is defined for all real numbers. This means that no matter what real number you substitute into the function for \( x \), the operation is valid and will yield a real number as an output.

It’s important to understand that with real numbers, you are dealing with an unbounded set. This is why in both the domain and range of cube root functions, we often see the notation (-∞, +∞), which implies that the function can take on and result in any real number without restrictions. Real numbers thus allow these functions to be flexible and continuous across the entire set of real numbers.