Problem 22 - Calculus Early Transcendentals

Question

Sketch the graph of the function. $$f(x, y)=y$$

Step-by-Step Solution

Answer

The graph of \(f(x,y) = y\) is a horizontal plane in 3D.

Step 1: Understand the Function

The function given is \(f(x,y) = y\). This means the function simply assigns the value of \(y\) to each pair \((x, y)\).

Step 2: Identify the Range and Domain

The function \(f(x, y) = y\) has a domain of all real numbers for both \(x\) and \(y\), i.e., \(x \in \mathbb{R}\) and \(y \in \mathbb{R}\). The range is also all real numbers, as \(f(x, y)\) outputs \(y\) directly.

Step 3: Visualize the Graph

The graph of \(f(x, y) = y\) in three-dimensional space is a horizontal plane. For every value of \(x\), the function output is exactly \(y\). There is no variation of the \(f(x, y)\) value with different \(x\) values; only the \(y\) value matters.

Step 4: Sketch the Graph

On a 3D coordinate system, draw a plane parallel to the \((xz)\)-plane. For each point \((x, y)\), you plot \((x, y, y)\) in the 3D space. Imagine a flat sheet extending infinitely along the \(x\)-axis and \(y\)-axis, without any slope.

Step 5: Examine the Characteristics

Since the graph is a plane, it's important to note the function's symmetry. The function is linear and simplifies the relationship between \(y\) and \(f(x,y)\). Every slice parallel to the \(xz\)-plane at a certain \(y = c\) will be a line \(z = c\).

Key Concepts

Function Visualization

Visualizing a function helps us understand its behavior and how it relates to different variables. For the function \( f(x, y) = y \), visualization is crucial because it involves mapping how the value of \( y \) remains consistent depending on the function output, regardless of \( x \).
In this case, when you plot the graph in three dimensions, you will see a plane. This plane reflects that every point on it matches the condition \((x, y, y)\). The function \( f(x, y) = y \) assigns each input \( (x, y) \) directly to \( y \). There is no dependency of the function's values on \( x \).
This kind of visualization emphasizes the uniform nature of the function regarding \( y \). It allows students to see how a mathematically abstract function can look like a tangible plane in space, helping bridge the gap between formula and form.

Domain and Range

When dealing with functions, it's essential to understand their domain and range. These concepts tell us what values a function can input and output.
The domain of a function represents all the possible input values. For \( f(x, y) = y \), the function inputs are the axis values \( x \) and \( y \). Since both are unrestricted, their domain is all real numbers: \( x \in \mathbb{R} \) and \( y \in \mathbb{R} \).

• Domain: All possible \( x \) and \( y \) values, making it \( \mathbb{R}^2 \).

The range tells us all the possible output values a function can have. Here, since the function outputs \( y \) itself, the range is simply all real numbers that \( y \) can take. Therefore, the range is also all real numbers:

• Range: \( y \in \mathbb{R} \).

Understanding domain and range is vital for predicting and confirming function behavior.

3D Coordinate Systems

Understanding 3D coordinate systems is essential for plotting functions like \( f(x, y) = y \). A 3D space adds depth to a conventional 2D view by introducing another axis, typically called the \( z \)-axis.
A point in this system is represented by three values: \((x, y, z)\). The \( x \)-axis, \( y \)-axis, and \( z \)-axis are mutually perpendicular, intersecting at the origin \((0, 0, 0)\).

• Visualization: The graph in our scenario is a plane in 3D space, plotting each point as \((x, y, y)\).
• Orientation: This graph is parallel to the \( xz \)-plane, meaning it extends infinitely along the \( x \) and \( y \) directions.

Grasping how these axes work together is crucial in drawing accurate representations of mathematical functions. It translates complex equations into visual elements, aiding deeper comprehension.

Linear Functions

Linear functions are characterized by their straight-line graph nature, even in 3D representations.
For the function \( f(x, y) = y \), it's a special case where the function represents a plane rather than just a line, due to the presence of two variables. This makes it a linear function because it satisfies the principle of superposition: doubling inputs doubles outputs.
This function maintains a consistent rate of change, here understood as flat, as it doesn't vary with \( x \). This creates:

• A straight-line slice when observed in any cross-section parallel to the axis planes.
• A comprehensive symmetry, seen when intersecting the \( xz \)-plane at any value of \( y \).

Slope, an important aspect of linear functions, here means there's no change in \( z \) with respect to \( x \), leading to a horizontal surface. This recognition helps clarify how linearity manifests in higher dimensions.