In Exercises 11–26, determine whether each equation defines y as a function of x. y²= x

Verified step by step guidance

1. Step 1: Recall the definition of a function. A function is a relation 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 each x-value produces a unique y-value.
2. Step 2: Analyze the given equation: y² = x. To isolate y, take the square root of both sides. This gives y = ±√x, meaning y can be either the positive or negative square root of x.
3. Step 3: Observe the result from Step 2. For any positive value of x, there are two possible values for y (one positive and one negative). This violates the definition of a function, as a single x-value corresponds to more than one y-value.
4. Step 4: Consider the special case when x = 0. Substituting x = 0 into the equation y² = x gives y² = 0, which simplifies to y = 0. In this case, there is only one value for y, but this does not change the overall conclusion for the equation.
5. Step 5: Conclude that the equation y² = x does not define y as a function of x because for most x-values, there are two corresponding y-values (positive and negative square roots).

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 y as a function of x.

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 more than once, then the relation is not a function. This test helps to quickly assess whether an equation like y² = x defines y as a function of x.

Implicit Functions

An implicit function is defined by an equation that relates the variables without explicitly solving for one variable in terms of the other. In the case of y² = x, y is not isolated, which means we need to consider both positive and negative roots when solving for y. This duality can affect whether y can be considered a function of x.