Problem 74 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The function \(f(x) = x\) is odd.

Step 1: Understand the Definitions

Even functions satisfy the condition \(f(-x) = f(x)\) for all \(x\). Odd functions satisfy \(f(-x) = -f(x)\) for all \(x\). To determine if \(f(x) = x\) is even or odd, we need to check these conditions.

Step 2: Evaluate the Expression with \(-x\)

Substitute \(-x\) into the function: \(f(-x) = -x\).

Step 3: Compare with Original Function

Compare \(f(-x)\) with \(f(x)\): \(f(-x) = -x\) and \(f(x) = x\). They are not equal, so \(f(x)\) is not even.

Step 4: Check Condition for Odd Function

Check if \(f(-x) = -f(x)\): Since \(f(-x) = -x\) and \(-f(x) = -x\), we have \(f(-x) = -f(x)\), indicating that the function is odd.

Key Concepts

even function

An even function is characterized by its symmetric behavior relative to the y-axis. This symmetry means that for any value of \(x\), when you substitute \(-x\) into the function, you get the same result as if you had used \(x\). Mathematically, this behavior is described as \(f(-x) = f(x)\) for all \(x\). For instance, functions such as \(f(x) = x^2\) and \(f(x) = \cos(x)\) are even because they fulfill this condition.

In simple terms, if you fold the graph of an even function along the y-axis, both halves will match perfectly. Recognizing an even function can be intuitive once you consider that the outputs change in the same way regardless of the sign of \(x\). This fundamental concept helps in evaluating and predicting the behavior of these functions within algebraic expressions.

function evaluation

Function evaluation involves determining the output of a function based on a particular input. Consider the function \(f(x) = x\), where each input value of \(x\) yields an output equal to that input. To evaluate \(f(-x)\), you substitute \(-x\) into the function wherever \(x\) appears.

For \(f(x) = x\), evaluating \(f(-x)\) would involve substituting to obtain \(f(-x) = -x\). The result \(-x\) indicates that the output for the input \(-x\) is simply the negative of \(x\). This process helps determine whether the function is odd or even by comparing these results to the original function.

algebraic expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the backbone of functions and provide a format for expressing mathematical ideas succinctly. For example, the expression \(f(x) = x\) is quite basic, displaying a direct relationship between the input \(x\) and the output.

In evaluating functions like \(f(x) = x\), simplicity lies in the directness of the expression. Algebraic expressions become particularly useful when checking symmetry in functions or determining changes that occur when inputs are modified, as when substituting \(-x\) in place of \(x\). This substitution process highlights the role of algebraic expressions in analyzing and interpreting functions.

symmetry in functions

Symmetry in functions is a key property used to categorize functions as even, odd, or neither. It defines how a function behaves when its input values are modified, specifically when substituting \(-x\) for \(x\). Two main types of symmetry exist:

• Y-Axis Symmetry: This is characteristic of even functions where \(f(-x) = f(x)\).
• Origin Symmetry: This is seen in odd functions where \(f(-x) = -f(x)\).

A function \(f(x) = x\), when compared to \(f(-x) = -x\), demonstrates origin symmetry, showcasing its odd nature. Understanding symmetry not only aids in determining whether a function is odd or even but also provides insights into the graph’s geometric characteristics. Recognizing symmetry in algebraic expressions helps in predicting and verifying behavior, offering a deeper understanding of the function’s properties.