Problem 56 - Calculus: One and Several Variables

Question

For \(x \geq 0 . f(x)=x^{2}-x .\) How is \(f\) defined for \(x < 0\) if (a) \(f\) is even? (b) \(f\) is odd?

Step-by-Step Solution

Answer

(a) For an even function, \(f(x)\) is defined as \(f(x)=x^{2}+x\) for \(x < 0\). (b) For an odd function, \(f(x)\) is defined as \(f(x) = -x^{2}-x\) for \(x < 0\).

Step 1 Find \(f\) for \(x < 0\) if \(f\) is even

For an even function, \(f(-x) = f(x)\). Substitute \(-x\) for \(x\) in \(f(x)\) to get the expression for \(f(-x)\). So, \(f(-x)=(-x)^{2}-(-x)=x^{2}+x\). For \(x < 0\), we must define \(f(x) = x^{2}+x\).

Step 2 Find \(f\) for \(x < 0\) if \(f\) is odd

For an odd function, \(f(-x) = -f(x)\). Substitute \(-x\) for \(x\) in \(f(x)\) to get the expression for \(f(-x)\). So, \(f(-x)=(-x)^{2}-(-x)=x^{2}+x\). We then change the sign to get \(-f(x)\) which gives us \(-x^{2}-x\). For \(x < 0\), we must define \(f(x) = -x^{2}-x\).

Key Concepts

Function Definition

In mathematics, a function is a specific relationship between a set of inputs and outputs. Each input corresponds to exactly one output, dictating how a function is defined. For example, the function \(f(x) = x^2 - x\) operates under the rule that for every \(x\), you substitute into the equation to get your result. The domain specifies the range of values \(x\) can take. For this problem, the function is initially defined as \(f(x) = x^2 - x\) for \(x \geq 0\). Understanding the definition of a function helps us explore other characteristics, such as how it behaves under different circumstances, such as negative values of \(x\).

Symmetry of Functions

Symmetry in functions refers to how functions mirror or replicate themselves about an axis. Even and odd symmetries are common types.

• **Even Functions**: They are symmetrical about the y-axis. This means that the function satisfies \(f(-x) = f(x)\) for all \(x\). An intuitive example is the function \(f(x) = x^2\), which looks the same on both sides of the y-axis.
• **Odd Functions**: These are symmetrical about the origin. This symmetry implies \(f(-x) = -f(x)\). A classic example is \(f(x) = x^3\), which rotates 180 degrees about the origin.

In the exercise, defining \(f\) for negative \(x\) involves using these symmetry rules. For an even function, \(f(-x)\) results in \(x^2 + x\). For an odd function, we determine that \(f(-x) = -x^2 - x\). This highlights the inherent symmetry properties of mathematical functions.

Polynomial Functions

Polynomial functions are significant in mathematics due to their wide array of applications and properties. They are expressions consisting of variables and coefficients that are combined using addition, subtraction, multiplication, and non-negative integer exponents.
Our function \(f(x) = x^2 - x\) is a polynomial of degree 2, commonly known as a quadratic polynomial.
Key features of polynomial functions include:

• **Degree**: The highest power of the variable in the expression. For \(f(x) = x^2 - x\), the degree is 2.
• **Coefficients**: The numbers that multiply the variables. Here, the coefficients are 1 and -1 for \(x^2\) and \(x\), respectively.
• **Roots**: Solutions to \(f(x) = 0\). For this polynomial, the roots can be found by solving \(x^2 - x = 0\).

Polynomial functions serve as foundational tools in algebra, calculus, and other mathematical disciplines.

Algebraic Manipulation

Algebraic manipulation involves rearranging algebraic equations to simplify or solve them. This is crucial when working with polynomial functions and determining how they behave for different conditions.
In the exercise, algebraic manipulation helps us find the expressions for \(f(x)\) and \(f(-x)\). This is done by substituting \(-x\) into the original function, leading to different algebraic expressions.
For example, after substitution in even functions, \(f(-x) = x^2 + x\). For odd functions, we adjust for symmetry, obtaining \(-f(-x) = -x^2 - x\).
Understanding these manipulations allows students to effectively handle more complicated functions and lays the groundwork for solving real-world problems.