The following equations implicitly define one or more functions.
b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Verified step by step guidance

1. Start by rewriting the given equation x + y³ - xy = 1 as y³ - 1 = xy - x. This sets up the equation for factoring.
2. Notice that the left side y³ - 1 is a difference of cubes, which can be factored using the formula a³ - b³ = (a - b)(a² + ab + b²). Here, a = y and b = 1.
3. Apply the difference of cubes formula to factor the left side: y³ - 1 = (y - 1)(y² + y + 1).
4. On the right side, factor out the common factor x from xy - x to get x(y - 1).
5. Set the factored forms equal: (y - 1)(y² + y + 1) = x(y - 1). To solve for y, consider the cases where y - 1 = 0 and where y - 1 ≠ 0, leading to different solutions for y.

Key Concepts

Implicit Functions

Implicit functions are defined by equations where the dependent variable is not isolated on one side. In this case, the equation x + y³ - xy = 1 defines y as a function of x without explicitly solving for y. Understanding how to manipulate these equations is crucial for identifying the relationships between variables.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of the given equation, factoring helps to simplify the equation after rewriting it, making it easier to isolate y and identify the functions defined by the equation.

Solving for y

Solving for y involves rearranging an equation to express y explicitly in terms of x. This is essential for identifying the functions y = f₁(x), y = f₂(x), etc. In the given problem, after rewriting and factoring the equation, one can isolate y to find the specific forms of the functions defined implicitly by the original equation.