Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x=y⁴

Verified step by step guidance

1. Identify the given relation: \(x = y^4\). Our goal is to determine if this relation defines \(y\) as a function of \(x\).
2. Recall that a relation defines \(y\) as a function of \(x\) if for every \(x\) value there is exactly one corresponding \(y\) value.
3. Try to solve the equation for \(y\) in terms of \(x\): \(y = \pm \sqrt[4]{x}\). Notice that for a given \(x\), there can be two values of \(y\) (positive and negative fourth roots), so \(y\) is not uniquely determined by \(x\).
4. Conclude that this relation does not define \(y\) as a function of \(x\) because each \(x\) (except possibly zero) corresponds to two \(y\) values.
5. Determine the domain and range: Since \(x = y^4\), \(x\) must be greater than or equal to zero (because any real number to the fourth power is non-negative), so the domain is \([0, \infty)\). The range of \(y\) is all real numbers \((-\infty, \infty)\) because \(y\) can be any real number.

Key Concepts

Definition of a Function

A function is a relation where each input x corresponds to exactly one output y. To determine if y is a function of x, each x-value must have only one y-value. If any x maps to multiple y-values, the relation is not a function.

Solving for y in Terms of x

To analyze whether y is a function of x, solve the given equation for y explicitly. For x = y^4, express y as y = ±x^(1/4). Since there are two possible y-values for each positive x, this indicates y is not a function of x.

Domain and Range of Relations

The domain is the set of all possible x-values, and the range is the set of all possible y-values. For x = y^4, the domain includes all x ≥ 0 (since y^4 ≥ 0), and the range includes all real y-values. Identifying these sets helps fully describe the relation.