Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 is even.

Step 1: Define Even and Odd Functions

A function is even if for every x in the domain, \( f(x) = f(-x) \). A function is odd if for every x in the domain, \( f(-x) = -f(x) \). Otherwise, the function is neither.

Step 2: Analyze the Given Function

The given function is \( f(x) = -4 \). This is a constant function, meaning it does not change with different values of \( x \). So, \( f(x) = -4 \) for any value of \( x \).

Step 3: Check for Even Function Property

Since \( f(x) = -4 \), for any \( x \), it follows that \( f(-x) = -4 \) as well. Therefore, \( f(x) = f(-x) \). This equality confirms that the function is even.

Step 4: Check for Odd Function Property

For the given function, \( f(-x) = -4 \) and \( -f(x) = -(-4) = 4 \). So \( f(-x) eq -f(x) \). The function does not satisfy the condition for being odd.

Step 5: Conclusion on Function Parity

Since the function satisfies the condition for being even, but not for being odd, \( f(x) = -4 \) is an even function.

Step 6: Sketching the Graph

The graph of \( f(x) = -4 \) is a horizontal line that passes through \( y = -4 \) on the y-axis. This line is parallel to the x-axis, confirming the even nature of the function as it is symmetrical about the y-axis.

Key Concepts

Parity of Functions

In mathematics, understanding whether a function is even or odd helps us comprehend its behavior and symmetry.

• An "even function" means that the function value is the same for both positive and negative inputs, i.e., for every number in the domain, the equation \( f(x) = f(-x) \) holds true.
• An "odd function" flips its value when we use the negative of the input, which means \( f(-x) = -f(x) \).
• When a function doesn't fulfill either condition, it is called neither even nor odd.

Even functions are symmetric about the y-axis, meaning they look the same on both sides of this axis. Recognizing function parity is crucial for understanding function behavior and can simplify graph sketching as well.

Constant Function

A constant function is one of the simplest types of functions. It has a neat characteristic: it outputs the same value no matter what input you put into it.
You can recognize a constant function by its form \( f(x) = c \), where \( c \) is a constant (a fixed number). In our exercise, we have \( f(x) = -4 \) which means it doesn't waver or fluctuate.

• No matter the x-value you choose, the y-value will always be \(-4\).
• This sets constant functions apart, as they're visually just horizontal lines on the graph, parallel to the x-axis.

Due to this consistency, constant functions are valuable in many mathematical applications where steady values are needed. But what makes them very intriguing is that these functions are automatically even since their value does not change regardless of the input's sign.

Graph Sketching

Sketching the graph of a function allows us to visually interpret the behavior and properties of that function.
For the constant function \( f(x) = -4 \), this graph is particularly simple. It is a straight, horizontal line that crosses the y-axis at \( y = -4 \).

• This line's parallel nature to the x-axis verifies its behavior as a constant function.
• Since it mirrors on both sides of the y-axis, it confirms its characteristic as an even function.

When plotting a constant function, remember:

• Identify the constant value \( c \).
• Draw a straight line at \( y = c \), with no slope.

This uncomplicated process of sketching not only solidifies the concept of constant functions but also provides a tangible way to see the symmetry inherent in even functions.