Problem 72 - College Algebra with Modeling and Visualization

Question

The table is a complete representation of \(f .\) Decide if \(f\) is even, odd, or neither. $$\begin{array}{rrrrrrr}x & -5 & -3 & -1 & 1 & 2 & 3 \\ f(x) & -4 & -2 & 1 & 1 & -2 & -4\end{array}$$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understanding Even and Odd Functions

An even function satisfies the condition \( f(x) = f(-x) \) for all \( x \) in the function's domain. An odd function satisfies \( f(x) = -f(-x) \) for all \( x \) in the domain.

Step 2: Checking the Even Function Condition

We need to check if \( f(x) = f(-x) \) for each value in the table. Compare the pairs: \( f(1) \) with \( f(-1) \), \( f(2) \) with \( f(-2) \), and \( f(3) \) with \( f(-3) \). Calculating, we find: \( f(1) = 1 \) and \( f(-1) = 1 \); \( f(2) = -2 \) and \( f(-2) \) does not exist, so we cannot check for x = 2 ; \( f(3) = -4 \) and \( f(-3) = -2 \). Thus, not all conditions for evenness hold.

Step 3: Checking the Odd Function Condition

Check if \( f(x) = -f(-x) \) for each value: \( f(1) = 1 \) and \( -f(-1) = -1 \); \( f(2) = -2 \) and \( f(-2) \) does not exist (so we cannot verify), indicating incompleteness from table for x = 2 ; \( f(3) = -4 \) and \( -f(-3) = 2 \). Not all conditions for oddness hold.

Step 4: Concluding if Neither

The function does not satisfy the conditions for being even or odd, since some pairs \( f(x) \) and \( f(-x) \) do not match either criterion consistently. Thus, function \( f \) is neither even nor odd.

Key Concepts

Function Symmetry

Function symmetry is a key concept when discussing even and odd functions and involves understanding how a function behaves when the inputs, or the "x-values," are mirrored around the origin or the y-axis. Think about it as a way of identifying whether a function creates a sort of mirror image.

• **Even Functions:** For an even function, reflecting across the y-axis doesn’t change its appearance. The defining characteristic is that for all inputs \( x \), \( f(x) = f(-x) \). You can imagine folding the graph on the y-axis, and if it aligns perfectly, it's probably even.
• **Odd Functions:** Odd functions, on the other hand, display rotational symmetry around the origin. This is akin to flipping the graph upside down around the origin. Mathematically, these functions satisfy \( f(x) = -f(-x) \).

Understanding these symmetries helps us make predictions about the behavior of functions without calculating every single output individually. In our exercise, by checking these symmetries and finding inconsistencies, we can see that the function is neither even nor odd.

Algebraic Functions

Algebraic functions are an essential part of mathematics, encompassing various expressions constructed from variables and mathematical operations like addition, subtraction, multiplication, division, and exponentiation.
These functions follow regular algebraic rules and can form polynomials, rational functions, and radicals, to name a few. Recognizing the structure of these functions lets us use algebraic properties for simplifying and solving problems.

• **Polynomials:** Easier to work with, polynomials involve expressions like \( ax^n + bx^{n-1} + ... + c \) where \( n \) is a non-negative integer. The nature of the coefficients and exponents plays a significant role in determining symmetry.
• **Function Symmetry in Polynomials:** Often, the exponents of polynomial terms determine symmetry. For instance, a polynomial with only even powers of \( x \) is an even function, and if it only has odd powers, it's an odd function.

In evaluating the symmetry of function \( f \) provided in the exercise, we observed it didn’t fit entirely within the neat categorization of being even or odd, suggesting a more complex algebraic behavior.

Function Evaluation

Function evaluation is the process of calculating the output of a function for a given input. It’s like plugging a number into a machine and seeing what result it spits out.
To evaluate a function means inserting specific x-values into the function's equation and determining the corresponding y-values (outputs).

• **Practical Steps:** Start by taking the given x-value, replace the variable in the function with this input, and simplify using algebraic rules to find \( f(x) \).
• **Multiple Evaluations:** Often, comparing different function evaluations helps identify patterns or symmetry, such as those seen in even or odd functions.

In our given exercise, evaluating \( f(x) \) at certain points helped us test the conditions for symmetry. By comparing \( f(x) \) with both \( f(-x) \) and \(-f(-x)\), we determined the lack of symmetry in the function, concluding it is neither even nor odd.