Problem 50 - Algebra 2

Question

Graph. Find the domain and the range of each function. \(y=4-\sqrt[3]{x+2.5}\)

Step-by-Step Solution

Answer

The domain of the function \(y=4-\sqrt[3]{x+2.5}\) is all real numbers, and the range of the function is also all real numbers. There is a horizontal shift of 2.5 units to the left, and a vertical shift of 4 units upwards.

Step 1: Identify the Domain

The domain of any cubic root function is all real numbers, because it is possible to take a cube root of any number, negative, positive or zero. This means that \(x\) can be chosen as any real number, so the domain of this function \(y=4-\sqrt[3]{x+2.5}\) is all real numbers.

Step 2: Identify the Range

The cubic root function \(y=\sqrt[3]{x}\) outputs all real numbers. The -4 outside the cubic root causes the entire graph to be shifted 4 units downwards, so the range (the set of possible values of \(y\)) of this function is also all real numbers.

Step 3: Identify any shifts

In this function \(y=4-\sqrt[3]{x+2.5}\), because \(x + 2.5\) is inside the cube root, this causes a 2.5 unit shift to the left. The +4 outside of the cube root causes a 4 unit upward shift. These shifts do not affect the domain or range of the function, but it's important to mentioned them.

Key Concepts

Domain

When discussing the domain of a function, we are asking what possible values can be input into the function for variable \(x\). For many functions, there are restrictions, such as not allowing for division by zero or taking square roots of negative numbers. However, when dealing with cube root functions, we have more flexibility.

• The cube root, denoted as \( \sqrt[3]{x} \), allows us to take the cube root of any real number: negative, positive, or zero.
• This is because cubing any real number (negative included) will yield a real result. Thus, there are no restrictions like you might find with square roots.
• For the function \( y=4-\sqrt[3]{x+2.5} \), the expression inside the cube root, \( x+2.5 \), does not affect the ability to take the cube root.

Therefore, the domain is all real numbers, represented as \((-\infty, \infty)\). Since the domain outlines the inputs, in this case, \(x\) can be any number without restriction.

Range

The range of a function refers to all possible output values \(y\), resulting from using the domain values. Let's delve into how the range is affected, especially by transformations within the function.

• For a standard cube root function \( y = \sqrt[3]{x} \), every real number can also be an output.
• While transformations, such as horizontal shifts (like \(x+2.5\)) or vertical shifts (indicated by the "+4") in our function, can change the placement of the graph, they do not restrict the range.
• The 4-unit upward shift moves all potential output values higher, but since we're dealing with \(y = 4 - \sqrt[3]{x+2.5}\), the cube root retains its ability to cover all real numbers.

Despite graph transformations, the nature of cube root functions ensures the range remains all real numbers, expressed as \([-\infty, \infty]\). This means any real \(y\) value is achievable by substituting some real number into \(x\).

Cube roots

Cube roots may initially appear challenging, but they are relatively straightforward compared to other root types. Let's simplify their understanding:

• The cube root of a number \(x\) is a value that, when cubed (raised to the power of three), gives \(x\).
• Expressed mathematically, \( \sqrt[3]{x} = y\) if and only if \(y^3 = x\).
• Unlike square roots, cube roots can be calculated for negative numbers, which is a distinct advantage. For instance, \( \sqrt[3]{-8} = -2\) because \((-2)^3 = -8\).

Understanding cube roots allows us to handle more complex functions like \(y=4-\sqrt[3]{x+2.5}\). We can confidently manage cube root functions since they inherently allow for a broader set of inputs and outputs without restriction.