Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-30, specify whether the given function is even, odd, or neither, and then sketch its graph. \(f(x)=-4\)

Step-by-Step Solution

Answer

The function \(f(x) = -4\) is even.

Step 1: Define an Even Function

A function is even if it satisfies the condition \(f(x) = f(-x)\) for all \(x\) in the domain of the function. This means the function is symmetric about the y-axis.

Step 2: Define an Odd Function

A function is odd if it satisfies the condition \(f(-x) = -f(x)\) for all \(x\) in the domain of the function. This means the function is symmetric about the origin.

Step 3: Check if the Function is Even

The given function is \(f(x) = -4\). Compute \(f(-x)\):\[f(-x) = -4\].Compare \(f(x)\) and \(f(-x)\):\[f(x) = -4\quad \text{and} \quad f(-x) = -4\].Since \(f(x) = f(-x)\), the function is even.

Step 4: Check if the Function is Odd

Since it was determined in Step 3 that \(f(x) = f(-x)\), the function cannot also satisfy the condition for being odd, which is \(f(-x) = -f(x)\). Therefore, the function is not odd.

Step 5: Conclusion about the Function

Since the function \(f(x) = -4\) satisfies the condition for being even, we classify it as an even function. It cannot be odd or neither, as it fits the definition of an even function.

Step 6: Sketch the Graph

The graph of the function \(f(x) = -4\) is a horizontal line at \(y = -4\). This means every point on the graph has the same y-coordinate, \(-4\), for any x-value. The graph is symmetric about the y-axis, confirming it is even.

Key Concepts

Function Symmetry

Understanding the symmetry of a function can simplify various mathematical analyses. A function exhibits symmetry when it maintains a consistent structure upon reflection. For even functions, symmetry is about the y-axis. This means if you take any point on the graph, the function value at this point and at the point directly opposite on the x-axis are equal. Mathematically, for a function \( f(x) \) to be even, it must satisfy \( f(x) = f(-x) \) for every \( x \) in its domain.
Odd functions, on the other hand, exhibit origin symmetry. This implies that rotating the function 180 degrees around the origin won't affect its structure. For an odd function, we must have \( f(-x) = -f(x) \) for every \( x \).
It's critical to determine whether a function is even or odd, as this greatly influences its behavior, aiding in other processes such as integration and prediction of function forms.

Graph Sketching

The ability to visualize a function's graph is a powerful tool in mathematics. Graph sketching involves drawing the function's visual representation, which helps understand its properties and behavior.
In the case of the function \( f(x) = -4 \), graph sketching is straightforward. Since it's a constant function, its graph is a horizontal line. Regardless of \( x \) values, the function returns \( -4 \) as output. Therefore, its graph is parallel to the x-axis and intersects the y-axis at \( y = -4 \).
When sketching graphs, consider symmetry. For \( f(x) = -4 \), observe it's symmetric about the y-axis, confirming its even nature. This symmetry helps ensure the graph's accuracy and further validates its classification as an even function.

Function Classification

Classifying functions based on their symmetry is essential for understanding their properties. Functions can be classified into three main categories: even, odd, or neither. This classification helps in predicting function behavior and simplifying operations like differentiation and integration.

• Even Functions: These exhibit y-axis symmetry. For example, the given function \( f(x) = -4 \) is an even function since \( f(x) = f(-x) \).


• Odd Functions: These have origin symmetry. An example would be any function satisfying \( f(-x) = -f(x) \).


• Neither: Functions that don't meet the criteria for even or odd fall into this category.

Recognizing these categories aids in efficiently navigating through more complex mathematical problems, leveraging the inherent properties of each classification.