Problem 49 - College Algebra

Question

For the following exercises, determine whether the function is odd, even, or neither. $$ h(x)=\frac{1}{x}+3 x $$

Step-by-Step Solution

Answer

The function \( h(x) = \frac{1}{x} + 3x \) is an odd function.

Step 1: Understanding Even and Odd Functions

A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \). A function is odd if \( f(-x) = -f(x) \) for all \( x \). If a function does not meet either of these conditions, it is neither even nor odd.

Step 2: Find \( h(-x) \)

Substitute \(-x\) in place of \(x\) in the function \( h(x) = \frac{1}{x} + 3x \).\[ h(-x) = \frac{1}{-x} + 3(-x) = -\frac{1}{x} - 3x \]

Step 3: Compare \( h(x) \) and \( h(-x) \)

We have \( h(x) = \frac{1}{x} + 3x \) and \( h(-x) = -\frac{1}{x} - 3x \). Comparing these:- \( h(-x) eq h(x) \) thus not even.- Check if \( h(-x) = -h(x) \):\[ -h(x) = -\left(\frac{1}{x} + 3x\right) = -\frac{1}{x} - 3x \]Since \( h(-x) = -h(x) \), the function is odd.

Key Concepts

Even and Odd Functions

In the world of algebra, understanding even and odd functions is crucial as it helps in analyzing and simplifying functions. To determine if a function is even or odd, you use specific criteria based on the function's behavior when the input variable, typically denoted as \(x\), is replaced by \(-x\). This replacement helps in observing the function's symmetry.

• **Even Functions:** A function \(f(x)\) is considered even if substituting \(-x\) gives the same result as \(f(x)\). Mathematically, this is shown as \(f(-x) = f(x)\). Graphically, even functions are symmetrical about the y-axis, meaning the left and right sides of the graph are mirror images.
• **Odd Functions:** A function is classified as odd if \(f(-x)\) gives the negative of \(f(x)\), represented as \(f(-x) = -f(x)\). Odd functions have a rotational symmetry about the origin, which means turning the graph 180 degrees around the origin will leave it unchanged.

If a function doesn't satisfy either condition, it is neither even nor odd.

Function Symmetry

Function symmetry is a helpful concept to visualize and understand how a function behaves graphically. There are several types of symmetry, but the two most relevant when discussing even and odd functions are y-axis symmetry and origin symmetry.

• **Y-Axis Symmetry:** When a function is mirrored across the y-axis, it is said to have y-axis symmetry. If a function is even, it will exhibit this type of symmetry. Looking at its graph, each point \((x, y)\) has a corresponding point \((-x, y)\).
• **Origin Symmetry:** This type of symmetry occurs when a graph can be rotated 180 degrees about the origin and remain unchanged, a trait of odd functions. In this case, every point \((x, y)\) mirrors to \((-x, -y)\) on the graph.

Recognizing these patterns assists in quickly identifying the function's properties and predicting its values without exhaustive calculations.

Algebraic Properties

Algebraic properties of functions involve analyzing the functional expressions themselves to determine their characteristics and how they change under different operations. Here, we explore how understanding these properties helps in identifying even and odd functions:

• Substitutions: To test if a function is even or odd, substitutions can be done algebraically as seen in the solution process where \(h(x) = \frac{1}{x} + 3x\) changes to \(h(-x) = -\frac{1}{x} - 3x\). Comparing these helps in classifying the function.
• Comparisons: Direct comparison between \(f(x)\) and \(f(-x)\) or comparing \(f(-x)\) to \(-f(x)\) can reveal function symmetry, helping in determining if a function is even, odd, or neither. If \(h(-x) = -h(x)\), the function is conclusively odd.

Grasping these algebraic techniques is a key skill in solving function-related problems efficiently and accurately.