Determine whether each equation defines y as a function of x. x = (1/3)(y^2)

Verified step by step guidance

1. Step 1: Recall the definition of a function. A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
2. Step 2: Consider the given equation: \( x = \frac{1}{3}y^2 \). We need to determine if for each x-value, there is only one corresponding y-value.
3. Step 3: Solve the equation for \( y \) in terms of \( x \). Multiply both sides by 3 to get \( 3x = y^2 \).
4. Step 4: Take the square root of both sides to solve for \( y \). This gives \( y = \pm \sqrt{3x} \).
5. Step 5: Analyze the result. Since \( y = \pm \sqrt{3x} \) gives two possible values of \( y \) for each positive \( x \), the equation does not define \( y \) as a function of \( x \).

Key Concepts

Function Definition

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. In mathematical terms, for a relation to be a function, no two ordered pairs can have the same first element with different second elements. This concept is crucial for determining if an equation defines one variable as a function of another.

Vertical Line Test

The vertical line test is a visual way to determine if a curve is a function. If any vertical line intersects the graph of the relation at more than one point, then the relation does not define y as a function of x. This test helps in understanding the graphical representation of functions and their properties.

Solving for y

To determine if an equation defines y as a function of x, it is often necessary to solve the equation for y. This involves isolating y on one side of the equation. If the resulting expression for y can be expressed as a single output for each input x, then y is a function of x; otherwise, it is not.