Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=|5 x|$$

Step-by-Step Solution

Answer

The function \( f(x) = |5x| \) is even.

Step 1: Understand Even and Odd Functions

A function is even if \( f(-x) = f(x) \) for all \( x \). A function is odd if \( f(-x) = -f(x) \) for all \( x \). We will use these definitions to evaluate if the given function \( f(x) = |5x| \) is even or odd.

Step 2: Compute \( f(-x) \)

Given \( f(x) = |5x| \), substitute \( -x \) into the function: \[ f(-x) = |5(-x)| = |-5x|. \] Since an absolute value is always positive, we simplify to \[ |-5x| = |5x|. \]

Step 3: Compare \( f(-x) \) with \( f(x) \)

Now that we have computed \( f(-x) = |5x| \) and we know \( f(x) = |5x| \), it's clear that \[ f(-x) = f(x). \] According to the definition, if \( f(-x) = f(x) \), then the function is even.

Step 4: Conclude the Function Type

Since \( f(-x) = f(x) \) for the function \( f(x) = |5x| \), we conclude that the function is even.

Key Concepts

Function Evaluation

Evaluating functions is a fundamental skill in mathematics.
It involves finding the value of a function for a specific input.
In simpler terms, you substitute the input value into the function to compute the result.
For example, if you have the function \( f(x) = |5x| \), to evaluate \( f(-x) \), replace \( x \) in the function with \( -x \).

• Start with the equation: \( f(x) = |5x| \)
• To find \( f(-x) \), substitute \( -x \) to get \( f(-x) = |5(-x)| = |-5x| \)
• Simplify, knowing absolute values always result in a positive value: \( |-5x| = |5x| \)

Evaluating functions like this helps determine properties, such as symmetry.
These properties lead to understanding whether a function is even or odd.

Absolute Value

The concept of absolute value is crucial for analyzing the given function.
The absolute value, denoted as \( |x| \), represents the distance of a number from zero on the number line. Thus, \( |x| \) is always non-negative.
For instance, both \( |5x| \) and \( |-5x| \) represent the non-negative magnitude of \( -5x \).
Absolute value has some core properties essential for understanding functions:

• It never outputs negative values: \( |-5| = 5 \), for example.
• \( |-a| = a \) if \( a \) is positive.
• \( |a| = |-a| \), meaning the absolute value is indifferent to the sign of \( a \).

When evaluating \( f(-x) = |-5x| \), you simplify it using these properties to determine \( |-5x| = |5x| \), leading to the conclusion about function symmetry.

Precalculus Concepts

Precalculus covers various essential mathematical concepts, including functions and their properties.
Understanding these basics prepares you for complex subjects like calculus.
One topic in precalculus is the classification of functions as even or odd, based on their behavior when evaluated at \(-x\).
Here’s a quick recap:

• An **even function** satisfies \( f(-x) = f(x) \). Examples include functions based on even powers (e.g., \( x^2 \)) and absolute values.
• An **odd function** satisfies \( f(-x) = -f(x) \). Odd powers often give rise to such behavior (e.g., \( x^3 \)).

In our exercise, recognizing \( f(x) = |5x| \) as even is straightforward because substituting \(-x\) yields the same output.
This stems from the property of absolute value that makes \( |-5x| = |5x| \).
These concepts are integral to precalculus and aid in understanding the broader structure of mathematical studies.