Problem 56 - A Graphical Approach to College Algebra

Question

Each function is either even or odd. Use \(f(-x)\) to state which situation applies. $$f(x)=-x^{5}+2 x^{3}-3 x$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Determine the Function's Nature

We need to determine whether the function \(f(x) = -x^5 + 2x^3 - 3x\) is even or odd. A function is even if \(f(-x) = f(x)\) and odd if \(f(-x) = -f(x)\).

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

Substitute \(-x\) for \(x\) in the function: \[f(-x) = -(-x)^5 + 2(-x)^3 - 3(-x)\] Simplify the expression: \[f(-x) = -(-x)^5 + 2(-x)^3 + 3x\] \[f(-x) = x^5 - 2x^3 + 3x\]

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

Calculate \(-f(x)\): \[-f(x) = -(-x^5 + 2x^3 - 3x) = x^5 - 2x^3 + 3x\] Compare \(f(-x) = x^5 - 2x^3 + 3x\) and \(-f(x) = x^5 - 2x^3 + 3x\). Since \(f(-x) = -f(x)\), the function is odd.

Key Concepts

even and odd functions

In mathematics, functions can be classified as either even or odd. A function is even if for every value of \(x\), \(f(-x) = f(x)\). Essentially, this means the function is symmetric with respect to the y-axis.An easy way to spot even functions is to look for symmetry in their graphs, such as the parabolas opening up or down.
On the other hand, a function is odd if \(f(-x) = -f(x)\), meaning the graph of the function is symmetric with respect to the origin.For example, an odd function would look the same after a 180-degree rotation around the origin.
Check whether a function is even or odd by substituting \(-x\) for \(x\) and comparing the results. Remember:

• An even function has y-axis symmetry.
• An odd function has origin symmetry.
• A function that is not even is either odd or neither.

polynomial functions

Polynomial functions play a crucial role when determining the nature of functions. They are expressions comprising terms in the form of \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer.For example, \(-x^5+2x^3-3x\) is a polynomial function of degree 5.
Polynomial functions are straightforward to analyze in terms of symmetry. You'll find patterns based on theexponents of the terms.

• Even-powered terms (like \(x^2\)) will contribute to the evenness of the function.
• Odd-powered terms (like \(x^3\)) will contribute to the oddness.
• The constant term doesn't affect the symmetry.

Understanding these patterns allows you to predict symmetry based on the polynomial structure.In our example \(-x^5+2x^3-3x\), each term has an odd power, indicating the function is likely odd.

function symmetry

Function symmetry is a fundamental concept to visually analyze and understand functions. Symmetrical properties can tell us how a function behaves or transforms.Function symmetry types include:

• Y-Axis Symmetry: Found in even functions where \(f(x) = f(-x)\). This means the left side of the graph mirrors the right side.
• Origin Symmetry: Characteristic of odd functions where \(f(x) = -f(-x)\). Here, the graph is mirrored both horizontally and vertically upon rotation.

Symmetry aids in the simplification of complex problems and in predicting a function’s graph without plotting too many points.When analyzing the function \(f(x)=-x^5+2x^3-3x\), calculating and comparing \(f(-x)\) and \(-f(x)\) showed origin symmetry, confirming that it’s an odd function.