Problem 11 - Precalculus with Calculus Previews

Question

Determine whether the given polynomial function \(f\) is even, odd, or neither even nor odd. Do not graph. \(f(x)=x^{5}+4 x^{3}+9 x+1\)

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understanding Even and Odd Functions

A function is even if its graph is symmetrical with respect to the y-axis, meaning for any value of x, \( f(x) = f(-x) \). A function is odd if its graph is symmetrical with respect to the origin, meaning \( f(-x) = -f(x) \). If neither condition is met, the function is neither even nor odd.

Step 2: Substitute and Simplify for f(-x)

Substitute \(-x\) for \(x\) in the function and simplify: \[ f(-x) = (-x)^5 + 4(-x)^3 + 9(-x) + 1 = -x^5 - 4x^3 - 9x + 1 \]

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

After simplifying \(f(-x)\), compare it with \(f(x)\). We have: \[ f(x) = x^5 + 4x^3 + 9x + 1 \] and \[ f(-x) = -x^5 - 4x^3 - 9x + 1 \] Since \(f(-x) eq f(x)\), the function is not even.

Step 4: Check for Odd Function Condition

To determine if the function is odd, check if \(f(-x) = -f(x)\). Find \(-f(x)\): \[ -f(x) = -(x^5 + 4x^3 + 9x + 1) = -x^5 - 4x^3 - 9x - 1 \] Since \(f(-x) eq -f(x)\), the function is also not odd.

Step 5: Conclusion

Since \(f(x)\) is neither equal to \(f(-x)\) nor equal to \(-f(x)\), the function \(f(x)\) is neither even nor odd.

Key Concepts

Even Functions

An even function is one where the function values do not change if the input is transformed into its opposite, meaning for any input \(x\), \(f(x) = f(-x)\). Imagine folding the graph of the function along the y-axis. If both halves overlap perfectly, then the function is considered even. Even functions exhibit symmetry with respect to the y-axis.

Classic examples of even functions include quadratic functions like \(f(x) = x^2\) and any polynomial with even powers, such as \(f(x) = x^4 + 2\).

When determining if a polynomial is even without graphing, simply substitute \(-x\) for \(x\) and simplify. Then compare \(f(-x)\) with \(f(x)\). If they are equal, you have an even function.
In the exercise provided, substituting \(-x\) gives \(f(-x) = -x^5 - 4x^3 - 9x + 1\), which is not the same as \(f(x) = x^5 + 4x^3 + 9x + 1\), showing this function is not even.

Odd Functions

Odd functions showcase a different type of symmetry, known as origin symmetry. This means if you rotate the graph 180 degrees around the origin, the graph looks the same. Mathematically, an odd function fulfills the condition \(f(-x) = -f(x)\).

This condition implies that for every positive value of \(x\), the negative counterpart, \(-x\), will yield a function value that is the exact opposite. Popular odd functions are those with only odd powers, like \(f(x) = x^3\) or linear functions such as \(f(x) = x\).

Checking oddness involves substituting \(-x\) into the function and simplifying to check if \(f(-x) = -f(x)\). In the exercise example, upon comparison, \(f(-x) = -x^5 - 4x^3 - 9x + 1\) is not equal to \(-f(x) = -x^5 - 4x^3 - 9x - 1\), indicating the function is not odd.

Symmetry of Functions

Symmetry in functions helps us understand their balanced behavior around an axis or a point. For polynomials, symmetry can be categorized into three types: y-axis symmetry (even functions), origin symmetry (odd functions), and asymmetrical functions which are neither even nor odd.

Even and odd symmetries offer significant hints about function behavior, especially in simplifying calculations or predicting function values. To detect symmetry:

• Check for y-axis symmetry by verifying if \(f(x) = f(-x)\).
• Check for origin symmetry by validating if \(f(-x) = -f(x)\).

If neither condition holds, the function does not exhibit significant symmetry. Instead, it is categorized as neither even nor odd, meaning the function does not mirror over an axis or rotate in any balanced way around the origin.
In the provided exercise, \(f(x)\) did not meet the criteria for either even or odd functions, thus it is classified as neither. Understanding these symmetries gives insights into function characteristics without necessarily graphing them.