Problem 75 - Algebra and Trigonometry

Question

\(75-82\) a Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ f(x)=x^{4} $$

Step-by-Step Solution

Answer

The function is even since \(f(x) = x^4\) satisfies \(f(-x) = f(x)\).

Step 1: Understand the Problem

Determine whether the function \(f(x) = x^4\) is even, odd, or neither. To do this, we need to evaluate the definition of even and odd functions.

Step 2: Recall Definitions

A function is **even** if \(f(-x) = f(x)\) for all \(x\). A function is **odd** if \(f(-x) = -f(x)\) for all \(x\). A function is **neither** if neither condition is satisfied.

Step 3: Evaluate f(-x) for x^4

Calculate \(f(-x)\) for the function \(f(x) = x^4\). This gives us \(f(-x) = (-x)^4 = x^4\), since raising \(-x\) to the fourth power results in \(x^4\).

Step 4: Compare f(x) and f(-x)

Compare \(f(x) = x^4\) with \(f(-x) = x^4\). Since \(f(x) = f(-x)\), no matter what \(x\) is, \(f(x)\) is an even function.

Step 5: Sketch the Graph Based on Symmetry

An even function is symmetric about the y-axis. Thus, when sketching the graph of \(f(x) = x^4\), it is symmetric to the left and right of the y-axis, forming a U-shaped curve similar to \(y = x^2\) but flatter at its minimum near zero.

Key Concepts

Function Symmetry

Symmetry in mathematics refers to an object or function that is invariant under certain transformations, such as reflections or rotations. For functions, symmetry is often detected by examining how the function behaves as inputs change in predictable ways.
For example, by substituting

• \(x\) with \(-x\)
• examining the output's relationship to the original function

we can recognize symmetrical properties in graphs. Specifically, if substituting \(x\) with \(-x\) within a function causes no change to the function's value, we say it exhibits even symmetry.
Conversely, if the resultant function is the negative of the original (i.e., \(f(-x) = -f(x)\)), it is considered to have odd symmetry. Recognizing these patterns allows us to predict and express the behavior of functions more efficiently without performing redundant computations.

Even Functions

Even functions are particularly interesting due to their symmetrical properties about the y-axis. The defining trait of an even function is that \(f(-x) = f(x)\) for every \(x\) in the domain of \(f\). This means that the graph of the function on the left half of the y-axis is a mirror image of the graph on the right half.
Common examples of even functions include polynomial functions with all even powers like \(f(x) = x^4\) or \(f(x) = x^2\). When verifying whether a function is even, substitute \(-x\) for \(x\) and check if the equation remains unchanged.

• If it does, the function is even.
• If it doesn't, the function is either odd or neither.

Recognizing even functions can be especially useful in various applications such as simplifying calculations and understanding the fundamental behavior of the function across its domain.

Graph Sketching

Graph sketching is an essential skill in visualizing and understanding functions. When sketching even functions, their symmetry about the y-axis means you can expect the graph to look identical on either side of this axis.
Consider the function \(f(x) = x^4\). Since it's even, start by plotting a few points for positive values of \(x\) and reflect these on the left side to easily create a complete sketch.
A practical approach is to:

• Choose specific values for \(x\).
• Compute \(f(x)\).
• Reflect these points across the y-axis to obtain additional points.

This method leverages the mirror-like quality of even functions. For \(f(x) = x^4\), you'll observe a U-shaped curve that is slightly flatter than \(f(x) = x^2\) at its base, and this demonstrates the characteristics of polynomial even-function behavior, giving a clear graphical representation of the function's properties.