Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y = x²

Verified step by step guidance

1. Step 1: Understand the definition of a function. A relation defines y as a function of x if for every x-value there is exactly one corresponding y-value.
2. Step 2: Analyze the given relation \(y = x^{2}\). For each value of \(x\), calculate \(y\) by squaring \(x\). Since squaring any real number \(x\) gives exactly one value of \(y\), this relation defines \(y\) as a function of \(x\).
3. Step 3: Determine the domain. The domain is the set of all possible input values \(x\) for which the function is defined. Since \(x\) can be any real number in \(y = x^{2}\), the domain is all real numbers, which can be written as \((-\infty, \infty)\).
4. Step 4: Determine the range. The range is the set of all possible output values \(y\). Since \(y = x^{2}\) is always greater than or equal to zero (because squaring any real number is non-negative), the range is \([0, \infty)\).
5. Step 5: Summarize: The relation \(y = x^{2}\) defines \(y\) as a function of \(x\) with domain \((-\infty, \infty)\) and range \([0, \infty)\).

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if y is a function of x, check that no x-value maps to multiple y-values. For y = x², each x has a unique y, so it defines y as a function of x.

Domain of a Function

The domain is the set of all possible input values (x-values) for which the function is defined. For y = x², since any real number can be squared, the domain is all real numbers, often written as (-∞, ∞).

Range of a Function

The range is the set of all possible output values (y-values) of the function. For y = x², since squaring any real number results in a non-negative value, the range is all real numbers greater than or equal to zero, written as [0, ∞).