Determine whether each equation defines y as a function of x. x+y³ = 8

Verified step by step guidance

1. Step 1: Recall the definition of a function. A function is a relationship where each input (x) corresponds to exactly one output (y). To determine if the given equation defines y as a function of x, we need to check if there is a unique value of y for every value of x.
2. Step 2: Start with the given equation: $x\; +\; y³\; =\; 8$. Rearrange the equation to isolate y³. Subtract x from both sides: $y³\; =\; 8\; -\; x$.
3. Step 3: To solve for y, take the cube root of both sides: $y\; =\; ∛(8\; -\; x)$. This step introduces the possibility of multiple values for y depending on the nature of the cube root.
4. Step 4: Recall that the cube root function is unique for real numbers. Unlike square roots, cube roots do not produce multiple values for a given input. Therefore, for every value of x, there is exactly one corresponding value of y.
5. Step 5: Conclude that the equation $x\; +\; y³\; =\; 8$ defines y as a function of x because the cube root operation ensures a unique output for each input.

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we must check if for every x, there is a unique y that satisfies the equation.

Vertical Line Test

The vertical line test is a visual method used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the relation is not a function, indicating that a single x-value corresponds to multiple y-values.

Implicit Functions

An implicit function is defined by an equation involving both x and y, where y is not isolated. In the equation x + y³ = 8, we need to analyze whether y can be expressed solely in terms of x, which affects whether it can be classified as a function.