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 equation y = x^2. For each x-value, calculate the corresponding y-value. Notice that for any real number x, there is only one y-value, which is x squared.
3. Step 3: Conclude that y = x^2 is a function because each x-value has a unique y-value.
4. Step 4: Determine the domain of the function. Since x can be any real number, the domain is all real numbers, expressed as (-∞, ∞).
5. Step 5: Determine the range of the function. Since y = x^2 is always non-negative, the range is all non-negative real numbers, expressed as [0, ∞).

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). In the context of the equation y = x², for every value of x, there is a unique value of y, making it a function. Understanding this definition is crucial for determining if a relation qualifies as a function.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). For the function y = x², the domain is all real numbers, as any real number can be squared. The range, however, is limited to non-negative real numbers since squaring any real number cannot yield a negative result.

Graphing Quadratic Functions

Quadratic functions, like y = x², produce a parabolic graph that opens upwards. This visual representation helps in understanding the behavior of the function, including its vertex, axis of symmetry, and intercepts. Graphing aids in identifying the domain and range, as well as confirming that the relation is indeed a function.