Question

Which of the following sets of ordered pairs \((x, y)\) illustrate why the equation \(y^2=x\) does not represent a function? A. \((4,2)\) and \((4,-2)\) B. \((9,3)\) and \((4,2)\) C. \((16,4)\) and \((9,3)\) D. \((4,2)\) and \((16,4)\)

Short answer

Option A, \( (4,2) \) and \( (4,-2) \), demonstrates why \( y^2 = x \) is not a function.

Step-by-step solution

Identify the definition of a function

A function is a relation where each input (x-value) is associated with exactly one output (y-value). If one x-value corresponds to more than one y-value, the relation is not a function.

Analyze each set of ordered pairs

Examine each provided pair of ordered pairs to determine if any x-value has more than one corresponding y-value.

Option A: \( (4,2) \) and \( (4,-2) \)

Both ordered pairs have the same x-value (4) but different y-values (2 and -2). This demonstrates the relation is not a function because one x-value has two different y-values.

Option B: \( (9,3) \) and \( (4,2) \)

Each ordered pair has a distinct x-value (9 and 4), which means there is no x-value associated with more than one y-value. This relation still satisfies the requirement of a function.

Option C: \( (16,4) \) and \( (9,3) \)

Similarly, each of these ordered pairs has a distinct x-value (16 and 9), and thus do not violate the definition of a function.

Option D: \( (4,2) \) and \( (16,4) \)

Each ordered pair has different x-values (4 and 16), so this also represents a valid function.

Conclusion

Out of the given options, only option A has a set where an x-value is associated with more than one y-value.

Key concepts

ordered pairs

Ordered pairs are a fundamental concept in mathematics, especially in functions and relations. They are written as \((x, y)\). Here, \((x)\) is the first element, referred to as the input or x-value, while \((y)\) is the second element, known as the output or y-value. Each ordered pair represents a specific point on a coordinate plane.

For instance, consider the pair \((4, 2)\). This means you move 4 units along the x-axis and 2 units along the y-axis to find the point. When analyzing ordered pairs, we focus on how these \((x)\) and \((y)\) values relate to define a function.

Ordered pairs help visually represent relationships between inputs and outputs. If each input is associated with exactly one output, then it forms a function.

input-output relationship

The input-output relationship is crucial in understanding functions. In mathematical terms, a function maps each input (x-value) to a specific output (y-value). This is akin to a machine where you feed it an input, and it gives a unique output.

If a machine generates multiple outputs for a single input, it cannot be considered a proper function. In the context of the exercise, we see this principle in action.

Look at option A: \((4, 2)\) and \((4, -2)\). Both pairs have the same x-value (4) but different y-values (2 and -2). This breaks the rule of functions since a specific input (4) maps to two different outputs. A proper function would have one unique output for each input.

However, in options B, C, and D, each input maps to a unique output, maintaining the function's integrity. Such an ordered structure helps in identifying and understanding functions clearly.

x-value and y-value

The x-value and y-value in ordered pairs play distinct roles in defining a function. The x-value, or the independent variable, is what we control or input into the function. The y-value, or the dependent variable, is what we get back as output.

To be classified as a function, each x-value must correspond to one and only one y-value. If an x-value relates to multiple y-values, the relationship is not a function. In mathematical notation, this is evident from the vertical line test. If a vertical line intersects a graph at more than one point, then the graph does not represent a function.

Let's revisit the given options. In option A, the x-value of 4 relates to multiple y-values (2 and -2), so it fails the vertical line test and, thus, isn't a function. Options B, C, and D pass this test as each unique x-value corresponds to only one y-value, confirming their status as functions.

Understanding x-values and y-values is integral in determining the nature of relationships and differentiating between functions and non-functions.