Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{1}{2 x}$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand the Definitions

A function is **even** if for all numbers \(x\) in the domain of \(f\), \(f(-x) = f(x)\). It is **odd** if \(f(-x) = -f(x)\). Our task is to determine which of these properties applies to the function \(f(x) = \frac{1}{2x}\). If neither holds, then the function is neither even nor odd.

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

We start by substituting \(-x\) into the function. \[ f(-x) = \frac{1}{2(-x)} = \frac{-1}{2x} \]

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

Now compare \(f(-x) = \frac{-1}{2x}\) to \(f(x) = \frac{1}{2x}\).It follows that:- \(f(-x) eq f(x)\), so the function is not even.- \(f(-x) = -f(x)\) since \(\frac{-1}{2x} = -\frac{1}{2x}\), so the function is odd.

Step 4: Conclusion: Determine Function Type

Since \(f(-x) = -f(x)\) is true, the function \(f(x) = \frac{1}{2x}\) is an odd function.

Key Concepts

Understanding Function Evaluation

Function evaluation involves substituting a specific input value into a function to find the output. In our exercise, we need to evaluate the function at \(-x\) to check if it's even or odd.
We began with the function \(f(x) = \frac{1}{2x}\). To evaluate \(f(-x)\), substitute \(-x\) into the function in place of \(x\):

• Original function: \(f(x) = \frac{1}{2x}\)
• Evaluated function: \(f(-x) = \frac{1}{2(-x)} = \frac{-1}{2x}\)

With this substitution, you can see how the output changes when the input is \(-x\) instead of \(x\). This evaluation is crucial because it allows us to compare outputs to determine properties like whether the function is even or odd.

Exploring Function Properties

Functions have various properties, including symmetry. Symmetry helps identify if a function is even or odd.
Understanding whether a function is even or odd gives insight into its behavior:

• A function is **even** if \(f(-x) = f(x)\) for all \(x\) in the domain. Graphically, it means the function is symmetrical about the y-axis.
• A function is **odd** if \(f(-x) = -f(x)\). This gives the graph rotational symmetry of 180 degrees about the origin.
• If neither condition is met, the function is neither even nor odd.

For \(f(x) = \frac{1}{2x}\), we evaluated \(f(-x) = \frac{-1}{2x}\) and compared it to \(f(x)\):

• \(f(-x) eq f(x)\), so \(f(x)\) is not even.
• \(f(-x) = -f(x)\), confirming that \(f(x)\) is odd.

These comparisons of \(f(x)\) and \(f(-x)\) highlight the function’s property as being odd.

Relating Precalculus Concepts to Even and Odd Functions

In precalculus, examining even and odd functions helps students understand function behavior deeply.
These concepts form a foundation for more complex mathematical ideas:

• Even functions simplify calculus work as they often have predictable derivatives and integrals.
• Odd functions provide interesting characteristics to their calculus when analyzing areas under curves or rotating around axes.
• The familiarity with even and odd functions aids in exploring polynomial, trigonometric, and other advanced functions later in calculus.

The function \(f(x) = \frac{1}{2x}\) demonstrates an odd function, revealing how such functions transform in a symmetric fashion around the origin. Recognizing these properties in precalculus builds confidence, preparing students for the intricacies of calculus.