Problem 28 - Calculus for Biology and Medicine

Question

Show that (a) \(y=x^{n}, x \in \mathbf{R}\), is an even function when \(n\) is an even integer. (b) \(y=x^{n}, x \in \mathbf{R}\), is an odd function when \(n\) is an odd integer.

Step-by-Step Solution

Answer

For even \( n \), \( y = x^n \) is an even function. For odd \( n \), \( y = x^n \) is an odd function.

Step 1: Understand Even and Odd Functions

An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain, while an odd function satisfies \( f(-x) = -f(x) \). We will use these definitions to demonstrate the properties of the function \( y = x^n \) for different values of \( n \).

Step 2: Prove Even Function for Even n

Let \( n \) be an even integer. Consider the function \( y = x^n \). For even \( n \), calculate \( y(-x) = (-x)^n = ((-1) \, x)^n = (-1)^n x^n = x^n \) because \( (-1)^n = 1 \) for even \( n \). Thus, \( y(-x) = y(x) \), confirming that the function is even.

Step 3: Prove Odd Function for Odd n

Let \( n \) be an odd integer. Consider the function \( y = x^n \). For odd \( n \), calculate \( y(-x) = (-x)^n = (-1)^n x^n = -x^n \) because \( (-1)^n = -1 \) for odd \( n \). Thus, \( y(-x) = -y(x) \), confirming that the function is odd.

Key Concepts

Even Function

An even function is one where the output of the function remains unchanged when the input is replaced by its negative counterpart. This characteristic is symbolically expressed as:

• \( f(-x) = f(x) \) for all \( x \) in the function's domain.

To understand this better, think of an even function as having symmetrical behavior around the y-axis in a graph.
This means if you fold the graph along the y-axis, the left and right sides would match perfectly.
A classic example of an even function is the quadratic expression like \( f(x) = x^2 \).

• For example, if \( f(x) = x^2 \), then \( f(-x) = (-x)^2 = x^2 \).
• Since \( f(-x) = f(x) \), the function is even.

Even functions often include polynomials with all even powers, like \( x^4 \), \( x^6 \), etc. Understanding this symmetry can help in identifying even functions quickly.

Odd Function

An odd function has a unique symmetry property, which is expressed in the equation:

• \( f(-x) = -f(x) \) for all values of \( x \) in the function's domain.

This equation implies that odd functions have rotational symmetry around the origin of the graph (0,0).
In simpler terms, if you rotate the graph 180 degrees around the origin, it will look the same.
An example of an odd function is the cubic function \( f(x) = x^3 \).

• When you replace \( x \) with \( -x \), you get \( f(-x) = (-x)^3 = -x^3 \).
• This shows \( f(-x) = -f(x) \), confirming that it's an odd function.

Odd functions generally include polynomials with all odd powers, like \( x^1 \), \( x^3 \), \( x^5 \), etc.
This rotational symmetry helps in recognizing odd functions in equations or graphical forms.

Integer Exponentiation

Integer exponentiation involves raising a number, known as the base, to an integer power, which is called the exponent.
The exponent tells you how many times to multiply the base by itself. Mathematically, it is represented as:

• \( x^n \)

where \( x \) is the base and \( n \) is the exponent. When \( n \) is a positive integer, you multiply the number by itself \( n \) times.

• For instance, \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \).

If \( n \) is a negative integer, it means you're dealing with the reciprocal of the base raised to the absolute value of the exponent.

• For example, \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).

Integer exponentiation forms the backbone of many mathematical functions and is key in defining even or odd functions depending on whether the integer exponent is even or odd.

Real Numbers

Real numbers encompass a wide range of numbers that include rational numbers, like fractions and integers, and irrational numbers, which cannot be expressed as simple fractions.
These are the numbers most frequently used in everyday mathematics, represented along the continuous number line.

• Rational numbers include integers such as -1, 0, 3, and fractions like \( \frac{1}{2} \), and \( 5.75 \).
• Irrational numbers are those like \( \pi \) and \( \sqrt{2} \), numbers that have non-repeating decimal points.

Real numbers are critical in many mathematics branches because they allow for the manipulation and calculation of any mathematical expression across continuous values. In the context of functions, analyzing behavior over the real numbers helps determine the nature of functions, such as deciding whether they are even or odd.