Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{x^{3}+3 x}{x}$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Calculate \(f(x)\)

First, simplify the given function. We have \(f(x) = \frac{x^3 + 3x}{x}\). Simplify by dividing each term in the numerator by \(x\). This gives us \(f(x) = x^2 + 3\).

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

Now, substitute \(-x\) into the simplified function. If \(f(x) = x^2 + 3\), then \(f(-x) = (-x)^2 + 3 = x^2 + 3\).

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

Check whether \(f(-x)\) matches \(f(x)\) to determine if it is even or odd. We find that \(f(-x) = x^2 + 3\) is the same as \(f(x) = x^2 + 3\).

Step 4: Determine if \(f(x)\) is even, odd, or neither

Since \(f(-x) = f(x)\), the function \(f(x)\) is even according to the definition of even functions.

Key Concepts

Understanding Function Evaluation

Function evaluation is a fundamental skill in algebra and calculus, allowing us to determine the output of a function for any given input. In our example, we have the function \(f(x) = \frac{x^3 + 3x}{x}\). To evaluate this, first simplify the function: divide each term in the numerator by \(x\), reducing it to \(f(x) = x^2 + 3\). This makes it easier to substitute values for \(x\) and understand how different inputs affect the output.

When we evaluate \(f\) at specific inputs like \(-x\), we're checking for patterns or properties such as symmetry. For this exercise, substituting \(-x\) into the function results in \((-x)^2 + 3\), which simplifies back to \(x^2 + 3\). Evaluating functions in this way helps to establish whether certain properties, like symmetry, apply to a given function.

Simplifying Expressions for Clarity

Simplifying expressions is crucial for understanding and working with functions effectively. In our function \(f(x) = \frac{x^3 + 3x}{x}\), simplifying involves breaking down the expression to its most basic form, \(x^2 + 3\).

This simplifies future calculations and evaluations, making it easier to substitute variables and to see any inherent properties of the function.

Here are some tips for simplifying functions effectively:

• Identify common factors in the numerator and denominator that allow for simplification.
• Break down each term separately if possible before simplifying.
• Reevaluate the expression to ensure that simplification has not changed the function's characteristics.

Simplifying expressions not only provides clarity but also aids in recognizing simpler patterns, such as symmetry, which might otherwise be obscured.

Exploring Symmetry in Functions

Symmetry in functions can reveal whether a function is even, odd, or neither. These characteristics can often simplify the process of understanding the behaviour of complex functions.

A function is even if substituting \(-x\) results in the same value as \(f(x)\). In our example, both \(f(x) = x^2 + 3\) and \(f(-x)\) evaluate to the same expression, indicating that \(f(x)\) is even.

Understanding symmetry:

• Even functions: These have symmetry about the y-axis. Graphically, this means that the left and right sides of the graph are mirror images.
• Odd functions: These have rotational symmetry about the origin. If \(f(-x) = -f(x)\), then the function is odd.
• Neither: If \(f(x)\) doesn't meet the criteria for either even or odd, it is neither.

Recognizing these symmetries simplifies graphing the function and understanding its properties more deeply. Symmetry also plays a fundamental role in higher-level mathematics, contributing to solving integrals and understanding series.