Problem 5 - College Algebra

Question

How can you determine whether a function is odd or even from the formula of the function?

Step-by-Step Solution

Answer

Substitute \(-x\) for \(x\) and compare \( f(-x) \) with \( f(x) \). If \( f(-x) = f(x) \), it's even; if \( f(-x) = -f(x) \), it's odd.

Step 1: Identify the Formula

Examine the given function formula. Let's denote the function as \( f(x) \). An example of a function could be \( f(x) = x^3 - x \).

Step 2: Substitute \(-x\) for \(x\)

Replace every occurrence of \( x \) in the function with \( -x \) to determine the form of \( f(-x) \). For example, if \( f(x) = x^3 - x \), then \( f(-x) = (-x)^3 - (-x) = -x^3 + x \).

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

Compare the expressions of \( f(x) \) and \( f(-x) \):- If \( f(-x) = f(x) \), the function is **even**.- If \( f(-x) = -f(x) \), the function is **odd**.

Step 4: Confirm the Function's Property

Based on the comparisons:- For the function \( f(x) = x^3 - x \), the calculated \( f(-x) = -x^3 + x \) is equal to \(-f(x)\), confirming that the function is odd.

Key Concepts

Function Symmetry

Function symmetry plays a critical role in understanding the nature of functions. It primarily revolves around two types: even and odd functions. These terms describe how a function behaves when its input is negated.

• An **even function** is symmetrical with respect to the y-axis. Mathematically, it means that substituting every occurrence of \( x \) with \( -x \) results in the same function, or \( f(-x) = f(x) \). A classic example of this is \( f(x) = x^2 \), where the graph will look identical on both sides of the y-axis.
• An **odd function** shows symmetry with respect to the origin. This means that when you substitute \( -x \) for \( x \), the function becomes the negative of the initial function, \( f(-x) = -f(x) \). An example of an odd function is \( f(x) = x^3 \), where rotating the graph 180° around the origin leaves it unchanged.

Understanding these symmetries allows you to predict and analyze the function's graphical behavior with greater insight.

Algebraic Functions

Algebraic functions are constructed using algebraic operations such as addition, subtraction, multiplication, division, and taking roots. They present a wide range of behaviors and symmetries.

• **Even functions in algebra** often contain even powers and constant components. A function like \( f(x) = x^4 + 2 \) is even because each term is unaffected by negating \( x \).
• **Odd functions in algebra** include terms with odd powers. An example is \( f(x) = x^3 - x \), recognized as odd because substituting \( -x \) changes the sign of every term.

By understanding these components, one can determine easily if a function is even, odd, or neither purely based on its algebraic formula.

Polynomial Functions

Polynomial functions are a subset of algebraic functions and consist of terms that are non-negative integer powers of \( x \). They can be easily analyzed for symmetry to determine if they are odd or even.

• For a **polynomial to be even**, each term must have an even exponent. Consider \( f(x) = 3x^2 + 4 x^0 \), which remains unchanged when \( x \) becomes \( -x \).
• An **odd polynomial** contains only terms with odd exponents. For instance, \( f(x) = x^5 - x^3 \) will change sign when \( x \) is replaced by \( -x \), confirming its odd property.

Recognizing these patterns in polynomial functions allows for straightforward identification of their symmetrical properties.