Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. x = y⁶

Verified step by step guidance

1. Step 1: Understand the problem by identifying the given relation: $x=y^6$.
2. Step 2: Determine if $y$ is a function of $x$. A relation defines $y$ as a function of $x$ if for every $x$ there is exactly one $y$.
3. Step 3: Analyze the equation $x=y^6$. For a given $x$, solve for $y$ to see if there is a unique solution.
4. Step 4: Consider the domain of the relation. The domain is the set of all possible $x$ values. Since $y^6$ is always non-negative, $x$ must be non-negative.
5. Step 5: Consider the range of the relation. The range is the set of all possible $y$ values. Since $y^6$ can be any non-negative number, $y$ can be any real number.

Key Concepts

Function Definition

A relation defines y as a function of x if each input x corresponds to exactly one output y. In other words, for every value of x in the domain, there should be a unique value of y. This concept is crucial for determining whether a given relation meets the criteria of a function.

Domain and Range

The domain of a function is the set of all possible input values (x-values) that can be used without causing any mathematical inconsistencies, while the range is the set of all possible output values (y-values) that result from those inputs. Understanding the domain and range helps in analyzing the behavior of the function and its graphical representation.

Inverse Relations

In the context of the relation x = y⁶, it is important to consider the inverse relationship between x and y. This relation can be rewritten as y = x^(1/6), which indicates that for each positive x, there are multiple corresponding y values (both positive and negative). This characteristic affects whether y can be considered a function of x.