Problem 59 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=3 x^{5}-x^{3}+7 x$$

Step-by-Step Solution

Answer

The function is odd, since \(f(-x) = -f(x)\).

Step 1: Understand the Property of Even and Odd Functions

A function is even if for all x in its domain, \(f(-x) = f(x)\). A function is odd if for all x in its domain, \(f(-x) = -f(x)\). We'll use these properties to determine if the function \(f(x) = 3x^5 - x^3 + 7x\) is even or odd.

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

Substitute \(-x\) into the function \(f(x) = 3x^5 - x^3 + 7x\). This gives:\[f(-x) = 3(-x)^5 - (-x)^3 + 7(-x)\].Simplifying each term, we get:\[f(-x) = 3(-x^5) - (-x^3) - 7x = -3x^5 + x^3 - 7x.\]

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

The function \(f(-x) = -3x^5 + x^3 - 7x\) is the opposite of \(f(x) = 3x^5 - x^3 + 7x\).We observe that \(f(-x) = -f(x)\), which indicates that the function is odd.

Key Concepts

Function Properties

In mathematics, functions can exhibit unique properties which can make their analysis easier. Two key properties are being "even" and "odd." These terms aren't just about numbers; they define specific relationships within functions.

- **Even Functions:** A function is even if it produces the same result when inputting both \(x\) and \(-x\). Mathematically, this is expressed as \(f(-x) = f(x)\) for all values in the domain of the function. Graphically, even functions are symmetrical across the y-axis.

- **Odd Functions:** A function is odd if applying a negative sign to the input results in both the opposite sign and opposite output. This means \(f(-x) = -f(x)\). Odd functions display rotational symmetry around the origin point.

Understanding these properties can aid in graphing and comprehending function behavior effectively.

Calculating f(-x)

To determine if a function is even or odd, one must calculate \(f(-x)\). This is achieved by substituting \(-x\) into every instance of \(x\) in the function's equation. Let's take a practical example to see how this works.

In the provided function, \(f(x) = 3x^5 - x^3 + 7x\), replace \(x\) with \(-x\). The calculation progresses like this:

• Start with the new input: \(f(-x) = 3(-x)^5 - (-x)^3 + 7(-x)\)
• Calculate each term individually: \(3(-x)^5 = -3x^5\), \(-(-x)^3 = x^3\), and \(7(-x) = -7x\)
• Simplify to find: \(f(-x) = -3x^5 + x^3 - 7x\)

Once the new expression is simplified, comparing it to the original function can tell us if it's even, odd, or neither.

Polynomial Functions

Polynomial functions like \(f(x) = 3x^5 - x^3 + 7x\) come with their own distinct characteristics. These functions are sums of terms consisting of a variable raised to a non-negative integer power, multiplied by a coefficient.

In our exploration of whether a polynomial is even or odd, the degree of each term plays a crucial part.

• **Odd-Degree Terms:** When the degree of the term (the power of \(x\)) is odd, its sign will change if \(x\) is replaced by \(-x\). This means if the original term was positive, it will become negative, and vice versa.
• **Even-Degree Terms:** Conversely, even-degree terms remain the same when \(x\) is substituted with \(-x\).

In the function \(f(x) = 3x^5 - x^3 + 7x\), all terms contain odd-degree powers (5, 3, and 1), indicating that it's highly likely the calculated \(f(-x)\) would contrast with \(f(x)\), affirming that the function is odd.