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Chapter 3: Problem 27

\((x, y) \in R\) if \(x=y^{2}\)

Short Answer

Expert verified
Any ordered pair (x, y) where x is equal to the square of y, and x is a nonnegative Real number, will satisfy the given equation \( x = y^{2}\). Or in other terms, \((x, y) \in R\) if \(x=y^{2}\) for \(x \geq 0\).

Step by step solution

01

Understanding the Function

The function given here is \(f(y) = y^{2}\), called a quadratic function. This function takes a real number y and squares it to give x. The graph of this function is a curve opening upwards (for \(y \geq 0\)) and downwards (for \(y < 0\)). Our task is to find all the possible values of x and y that satisfy the equation in the Real numbers.
02

Finding the Value of x

For any given value of y, the corresponding x can be found by computing \(y^{2}\). Since the only restriction is that y must be a real number, y can be any positive number, zero or any negative number. The function \((y^{2})\) would then produce a nonnegative number for x.
03

Visual Verification

To verify that this understanding is correct, someone can plot the function on a graph. For every y value the corresponding x value will be at the point on the curve defined by the function \(f(y) = y^{2}\). The x-values will always be nonnegative as we cannot square a real number and obtain a negative result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Numbers
Real numbers are numerical values that include all the numbers along the number line. This includes both rational numbers (such as fractions and integers) and irrational numbers (which cannot be expressed as a simple fraction like \( rac{2}{3}\)). Real numbers cover:
  • Positive numbers (e.g., 1, 2.5, \(\sqrt{2}\), etc.)
  • Negative numbers (e.g., -1, -3.5)
  • Zero
In the context of the exercise, when we say \(x, y \in R\), it means that both x and y are real numbers. When we have functions like \(x = y^2\), y can be any real number, whether positive, negative, or zero. However, x, which comes from \(y^2\), will always be non-negative since squaring a number always results in a zero or positive value.
Graph of Functions
The graph of a quadratic function like \(f(y) = y^{2}\) is a parabolic curve. This specific shape is symmetric around the vertical axis, known as the axis of symmetry. Key characteristics of the graph of \( f(y) = y^{2} \) include:
  • It is a U-shaped curve, called a parabola.
  • The vertex, or the lowest point, is at the origin \( (0,0) \).
  • For positive y-values, as y increases, x increases more steeply because of the squaring.
  • Similarly, for negative y-values, as y decreases, x also increases in magnitude but remains non-negative, reflecting the symmetry.
This graph can be visualized by plotting several points on a Cartesian plane and observing the smooth curve that connects them. The shape demonstrates that as y moves away from zero in either direction, x becomes larger, illustrating how y dominates the behavior of x through the squaring process.
Relationship Between Variables
In the function \(x = y^{2}\), x is the dependent variable and y is the independent variable. The relationship between x and y is captured by the quadratic nature of the function:
  • The value of x solely depends on the value of y.
  • Since y is squared, even if y is negative, x remains positive due to the squaring operation.
  • This highlights the nonlinear relationship: small changes in y result in larger changes in x when y is large.
Understanding this relationship is crucial. Because of squaring, each positive y-value has a corresponding negative y-value that gives the same x. For instance, both y = 2 and y = -2 give x = 4. This illustrates that the same x-value can correlate with several y-values, some positive and some negative, further emphasizing the symmetry of quadratic functions.

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