Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=x^{6}-4 x^{4}+5$$

Step-by-Step Solution

Answer

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

Step 1: Evaluate f(-x)

Substitute -x for x in the function f(x) = x^6 - 4x^4 + 5. This gives us f(-x) = (-x)^6 - 4(-x)^4 + 5. Calculate each term: - (-x)^6 = x^6 because raising a negative number to an even power results in a positive number. - -4(-x)^4 = -4x^4 for the same reason as above. - The constant term 5 remains 5. Thus, f(-x) = x^6 - 4x^4 + 5.

Step 2: Compare f(-x) and f(x)

Compare the function f(-x) = x^6 - 4x^4 + 5 to f(x) = x^6 - 4x^4 + 5. Observe that f(-x) is exactly equal to f(x).

Step 3: Determine if the function is even or odd

Since f(-x) = f(x), the function is even. A function is considered even when substituting -x for x does not change the function, meaning it is symmetric about the y-axis.

Key Concepts

Function Evaluation

Function evaluation helps us understand the behavior of a function when various substitutions are made. In this context, we aim to evaluate the function by substituting \(x\) with its negative counterpart, \(-x\). This involves taking the given function \(f(x) = x^6 - 4x^4 + 5\) and substituting \(-x\):

• For the term \(x^6\), substituting gives \((-x)^6 = x^6\). Raising a negative number to an even power results in a positive outcome, leaving this term unchanged.
• Similarly, for the term \(-4x^4\), we find \((-x)^4 = x^4\), leading to \(-4(-x)^4 = -4x^4\).
• The constant term \(5\) remains unaffected because it is not associated with the variable \(x\).

Combining these evaluations, we conclude that \(f(-x) = x^6 - 4x^4 + 5\), exactly mirroring the original function. Understanding these steps is crucial for determining the symmetry and classification of the function as even or odd.

Symmetry in Functions

Symmetry in functions provides insight into how functions are mirrored across axes. For even and odd functions, symmetry helps identify their specific characteristics:

• An even function maintains symmetry about the y-axis. This means that for any input \(x\), the function's output remains unchanged when \(x\) is replaced by \(-x\). Mathematically, this is represented as \(f(-x) = f(x)\).
• In contrast, an odd function is symmetric about the origin, satisfying the condition \(f(-x) = -f(x)\). This implies rotational symmetry of 180 degrees around the origin.

In evaluating the given function, \(f(x) = x^6 - 4x^4 + 5\), it is confirmed that \(f(-x) = f(x)\), thus classifying the function as even. The plot of such a function will visually depict this symmetry as it mirrors evenly across the y-axis.

Polynomial Functions

Polynomial functions are a standard class of functions comprising terms consisting of variables raised to non-negative integer powers. The general form of a polynomial function is given by \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). They are widely used in mathematical analysis for their diverse properties:

• The degree of a polynomial is determined by the highest power of the variable \(x\). In the case of \(f(x) = x^6 - 4x^4 + 5\), the degree is 6.
• Polynomial functions often display smooth and continuous curves in their graphs, with behavior influenced by their degree and leading coefficients.
• Understanding the symmetry of a polynomial is crucial, as it directly impacts graph interpretations and transformations.

The function given is an even polynomial, exemplifying how even degrees (here, 6) can suggest an even function, which aids in predicting graph behavior and symmetry characteristics.

Graphical Approach

The graphical approach to examining functions allows us to visualize their behaviors and symmetries. By plotting the function \(f(x) = x^6 - 4x^4 + 5\), several key features emerge:

• The graph of an even function appears symmetric about the y-axis. For this particular function, this symmetry implies that flipping the graph across the y-axis will overlay it perfectly onto itself.
• The shape and curvature of the function, influenced by the leading term \(x^6\), results in a smooth, continuous curve with specific peaks and troughs determined by other terms.
• Visualizing the function also aids in understanding points of intersection, turning points, and intervals of growth or decrease.

Through graphing, these insights heavily complement algebraic observations made through function evaluation and symmetry identification. This makes graphing an invaluable tool for comprehensively understanding even and odd characteristics of polynomial functions.