Problem 49 - College Algebra with Corequisite Support

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: Understand Definitions

A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain. It is odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \) in its domain. If neither condition is satisfied, the function is neither even nor odd.

Step 2: Substitute \( -x \) into the Function

Compute \( h(-x) \) to evaluate the symmetry. Substitute \( -x \) into \( h(x) = \frac{1}{x} + 3x \). This gives us:\[h(-x) = \frac{1}{-x} + 3(-x) = -\frac{1}{x} - 3x.\]

Step 3: Check If Function is Even

Recall the condition for even functions: \( h(-x) = h(x) \). We compare \( h(-x) = -\frac{1}{x} - 3x \) with \( h(x) = \frac{1}{x} + 3x \). Since \( -\frac{1}{x} - 3x eq \frac{1}{x} + 3x \), the function does not satisfy this condition, so it is not even.

Step 4: Check If Function is Odd

Recall the condition for odd functions: \( h(-x) = -h(x) \). Calculate \(-h(x)\) for comparison:\[-h(x) = -\left( \frac{1}{x} + 3x \right) = -\frac{1}{x} - 3x.\]Notice \( h(-x) = -\frac{1}{x} - 3x \) matches perfectly with \(-h(x)\). This means \( h(x) \) satisfies the condition to be an odd function.

Key Concepts

Odd Function

An odd function is a type of function in mathematics with a special kind of symmetry. Odd functions have rotational symmetry about the origin in a graph. To determine if a function is odd, you should check if the function satisfies the condition:

• \( f(-x) = -f(x) \) for all \( x \) in the domain.

Let's see why this symmetry happens:

• When you replace \( x \) with \( -x \) in the function, it flips or mirrors over the y-axis.
• Then, checking if this equals \(-f(x)\), the function should also reflect over the x-axis.

This double flipping indicates that the graph will match its original if rotated 180 degrees around the origin.
Therefore, when you find that \( h(-x) \) equals \(-h(x)\), like in our function \( h(x) = \frac{1}{x} + 3x \), the function is odd. In this function, substituting \( h(-x) = -\frac{1}{x} - 3x \) and comparing to \(-h(x)\) allows us to confirm its oddness.

Even Function

Even functions are characterized by symmetry about the y-axis. Here's how you can determine if a function is even:

• A function \( f(x) \) is even if it satisfies the condition \( f(-x) = f(x) \).

This symmetry means that each point on the right of the y-axis at \( x \) will have a corresponding point directly opposite on the left at \( -x \).
Let's break this down:

• Upon replacing \( x \) with \( -x \), the function should remain unchanged.
• This reflects that the function's graph mirrors itself perfectly across the y-axis.

In the original exercise function \( h(x) = \frac{1}{x} + 3x \), substituting \( -x \) yields \( -\frac{1}{x} - 3x \). We observed that this does not equal \( h(x) \), thus the function is not even as it fails to satisfy the condition \( f(-x) = f(x) \). It's only through such substitutions that we can clearly see the symmetry properties of even functions.

Algebraic Functions

Algebraic functions like \( h(x) = \frac{1}{x} + 3x \) are expressions made up of algebraic operations, such as addition, subtraction, multiplication, division, and taking roots. When examining symmetry, the structure of these functions is important.
Key characteristics of algebraic functions include:

• Their expressions involve constants and variables raised to rational powers.
• They can be complex, involving combinations of polynomials or rational expressions.

In symmetry analysis, recognizing if an algebraic function is even or odd comes down to substituting \(-x\) and analyzing its effect.

• For example, in our function, substituting \(-x\) into each term clearly shows how it impacts symmetry, helping identify whether the function is odd or even.

Algebraic functions are not predefined as even or odd; rather, each function must be individually examined and substituted to establish its symmetry characteristics. Understanding the interaction of operations within these functions guides us in determining their symmetry properties.