Problem 21 - Precalculus Essentials

Question

Determine whether each equation defines y as a function of \(x .\) $$x+y^{3}=8$$

Step-by-Step Solution

Answer

Yes, the equation \(x + y^{3} = 8\) defines \(y\) as a function of \(x\).

Step 1: Analyze the Equation

The given equation is \(x + y^{3} = 8\). For \(y\) to be a function of \(x\), each value of \(x\) must correspond to exactly one value of \(y\).

Step 2: Solve for \(y\)

Rearrange the equation to express \(y\) in terms of \(x\): \(y^{3} = 8 - x\). Then, take the cube root of both sides to solve for \(y\): \(y = \sqrt[3]{8 - x}\). This equation gives a unique value of \(y\) for each value of \(x\), hence, \(y\) is a function of \(x\).

Key Concepts

Determining if an Equation Defines a Function

An important part of understanding functions is determining whether a given equation defines a function. A function is a relationship where each input (often represented by \(x\)) is associated with exactly one output (often represented by \(y\)).
To determine if an equation defines \(y\) as a function of \(x\), we need to analyze whether every possible \(x\) value corresponds to one and only one \(y\) value.

• If any line you draw that is parallel to the y-axis intersects the graph of the equation more than once, the equation does not define \(y\) as a function of \(x\).
• This is known as the "vertical line test."

By examining the equation \(x + y^{3} = 8\), we can rearrange it to find \(y = \sqrt[3]{8 - x}\), showing that each \(x\) yields a unique \(y\) owing to the properties of the cube root function.

Solving Equations

Solving equations often involves isolating one of the variables to express it in terms of the other variables. This process generally requires algebraic manipulation.
For the equation \(x + y^{3} = 8\), we aimed to express \(y\) in terms of \(x\). This involved the following steps:

• Subtract \(x\) from both sides to get \(y^3 = 8 - x\).
• Take the cube root of both sides to solve for \(y\): \(y = \sqrt[3]{8 - x}\).

It is crucial to follow these steps carefully, as any error in the process might lead to incorrect conclusions about the relationship between \(x\) and \(y\).

Cube Root

The cube root is a valuable concept in mathematics, especially when dealing with equations involving powers of three. The cube root of a number \(a\), denoted as \(\sqrt[3]{a}\), is the value \(b\) such that \(b^3 = a\).
Unlike square roots which can sometimes yield two values (positive and negative), cube roots always produce a single real number value, even for negative numbers. This characteristic is useful in determining functions, as we see with \(y = \sqrt[3]{8 - x}\), ensuring a unique \(y\) value for every \(x\).

• The cube root is the inverse operation of raising a number to the third power.
• This operation is particularly straightforward because the cube of any real number covers the whole range of real numbers without repeating any values.

Function Notation

Function notation is a way to name functions and describe their output precisely. Instead of writing equations like \(y = \sqrt[3]{8 - x}\), we use notation like \(f(x)\) to indicate a function.

• Here, \(f(x)\) denotes the function's output when \(x\) is the input value.
• This notation helps in clearly identifying which variable is the independent variable (input), and which is the dependent variable (output).

When working with function notation, it simplifies expressing complex relationships, and facilitates operations such as adding, subtracting, or composing functions. Understanding this notation is essential for working effectively with functions in mathematics.