Problem 56 - Algebra 2

Question

Graph each relation or equation and find the domain and range. Then determine whether the relation or equation is a function. $$ x=y^{2} $$

Step-by-Step Solution

Answer

The relation is not a function. Domain: \([0, \infty)\); Range: \((-\infty, \infty)\).

Step 1: Write the Equation in Terms of y

The given equation is \( x = y^2 \). To find the expression for \( y \), solve for \( y \):\[ y = \pm \sqrt{x} \]This shows that for every positive \( x \), \( y \) has two possible values.

Step 2: Sketch the Graph

Plot the graph of \( y = \sqrt{x} \) and \( y = -\sqrt{x} \). These are parabolas that open to the right and represent both positive and negative values of \( y \) for every \( x \geq 0 \). The graph is symmetric around the x-axis.

Step 3: Determine the Domain

The domain of a relation is the set of all possible \( x \)-values. Since \( y = \pm \sqrt{x} \) is defined for \( x \geq 0 \), the domain is:\[ [0, \infty) \]

Step 4: Determine the Range

The range of a relation is the set of all possible \( y \)-values. Since \( y \) can take any real positive or negative values (and zero) as long as \( x \geq 0 \), the range is:\[ (-\infty, \infty) \]

Step 5: Determine if the Relation is a Function

A relation is a function if each input (\( x \)-value) corresponds to exactly one output (\( y \)-value). For each positive \( x \), there are two possible \( y \)-values (both positive and negative), which violates the definition of a function. Therefore, this equation is not a function.

Key Concepts

Domain and Range

When dealing with mathematical relations and functions, understanding the domain and range is essential. The **domain** refers to the set of all possible input values, often written as "x-values," that a relation or function can take. This is essentially your input space. For the equation \(x = y^2\), the domain includes all non-negative numbers \(x\geq 0\), because a square root of a negative number does not result in a real number. Thus, the domain is \([0, \infty)\).
Similarly, the **range** is the set of all possible output values, or "y-values," that can be derived from the domain. For our equation, since \(y = \pm \sqrt{x}\), all real numbers \(y\) are possible when \(x\) is zero or positive. This means the range is \( (-\infty, \infty) \), covering both positive and negative numbers as well as zero. Understanding these concepts helps clarify what values are permitted and what outputs can be expected when dealing with functions or relations in mathematics.

Graphing Relations

Graphing relations involves plotting all possible points that satisfy the equation or inequality on a coordinate plane. For the relation \(x = y^2\), visualize this by solving for \(y\), obtaining \(y = \pm \sqrt{x}\).
This leads us to creating two distinct graphs:

• \(y = \sqrt{x}\)
• \(y = -\sqrt{x}\)

These graphs form parabolas that open sideways to the right, one upwards for positive \(y\) values and one downwards for negative \(y\) values. This pattern creates a "mirror" image around the x-axis, and the vertex of these parabolas is located at the origin (0,0).
While plotting these points, remember that only non-negative values of \(x\) are valid due to the nature of the square root function. Consequently, the graph lies entirely on the right side of the y-axis.

Parabolas

A parabola is a symmetrical, curved shape that represents quadratic relations in algebra. In the equation \(x = y^2\), the presence of \(y^2\) signifies the characteristics of a parabola. Commonly, parabolas are seen in forms like \(y = ax^2\), opening upwards or downwards, but here we see a left-to-right orientation due to the form \(x = y^2\).
Key features of parabolas:

• They have a vertex, a point that is the minimum or maximum, depending on the orientation.
• They are symmetric about a line, which is often the axis on which they "open." For our sideways parabolas, the axis is the y-axis.
• They can represent real-world phenomena like projectile motion, which similarly follow parabolic paths.

Understanding how to manipulate the standard parabolic equation to fit specific graphical requirements broadens the scope of depicting various real-world and theoretical scenarios.