Power Functions (VCE SSCE Mathematical Methods): Revision Notes

Power Functions

Introduction

Power functions are mathematical functions with the form $f(x) = x^r$, where $r$ is a rational number. These functions appear frequently in mathematics and have different properties depending on the value and type of the exponent.

Examples of power functions include:

• $f(x) = x^4$ (even positive integer power)
• $f(x) = x^{-4}$ (even negative integer power)
• $f(x) = x^{\frac{1}{4}}$ (fractional power - fourth root)
• $f(x) = x^5$ (odd positive integer power)
• $f(x) = x^{-5}$ (odd negative integer power)
• $f(x) = x^{\frac{1}{3}}$ (fractional power - cube root)

Increasing and decreasing functions

Understanding whether a function is increasing or decreasing is fundamental to analysing power functions.

Strictly increasing functions

A function $f$ is strictly increasing on an interval when larger input values always produce larger output values. Mathematically, this means:

Examples of strictly increasing functions:

• Any straight line with a positive gradient
• The function $f: [0, \infty) \to \mathbb{R}, f(x) = x^2$ (on the non-negative domain)

Strictly decreasing functions

A function $f$ is strictly decreasing on an interval when larger input values always produce smaller output values. Mathematically, this means:

Examples of strictly decreasing functions:

• Any straight line with a negative gradient
• The function $f: (-\infty, 0] \to \mathbb{R}, f(x) = x^2$ (on the non-positive domain)

Power functions with positive integer index

When we have $f(x) = x^n$ where $n$ is a positive integer, the graph's shape depends critically on whether n is odd or even.

The simplest cases are familiar:

• $n = 1$: Linear function $f(x) = x$
• $n = 2$: Quadratic function $f(x) = x^2$
• $n = 3$: Cubic function $f(x) = x^3$

Functions with odd positive integer powers

For $f(x) = x^n$ where $n$ is an odd positive integer (such as $n = 3, 5, 7, \ldots$), the function has these important properties:

Domain and range:

• Maximal domain: All real numbers ($\mathbb{R}$)
• Range: All real numbers ($\mathbb{R}$)

Key properties:

• The function is odd, meaning it has rotational symmetry about the origin: $f(-x) = -f(x)$
• The function is strictly increasing across its entire domain
• The function is one-to-one (each output corresponds to exactly one input)
• Special values: $f(0) = 0$, $f(1) = 1$, and $f(-1) = -1$
• As $x \to \infty$, we have $f(x) \to \infty$
• As $x \to -\infty$, we have $f(x) \to -\infty$

Functions with even positive integer powers

For $f(x) = x^n$ where $n$ is an even positive integer (such as $n = 2, 4, 6, \ldots$), the function has these important properties:

Domain and range:

• Maximal domain: All real numbers ($\mathbb{R}$)
• Range: Non-negative real numbers ($\mathbb{R}^+ \cup \{0\}$)

Key properties:

• The function is even, meaning it has reflective symmetry about the y-axis: $f(-x) = f(x)$
• The function is strictly increasing for $x \geq 0$
• The function is strictly decreasing for $x \leq 0$
• Special values: $f(0) = 0$, $f(1) = 1$, and $f(-1) = 1$
• As $x \to \pm\infty$, we have $f(x) \to \infty$

The graphs create U-shaped parabolas. Higher even powers produce curves that are flatter near the origin and steeper further from the origin.

Power functions with negative integer index

Power functions with negative integer exponents are reciprocal functions. Their behaviour also depends on whether the index is odd or even.

Functions with odd negative integer powers

For $f(x) = x^n$ where $n$ is an odd negative integer, we can write $f(x) = x^{-k}$ for $k = 1, 3, 5, \ldots$

The simplest case is $f(x) = x^{-1} = \frac{1}{x}$, which creates a rectangular hyperbola.

Domain and range:

• Maximal domain: $\mathbb{R} \setminus \{0\}$ (all real numbers except zero)
• Range: $\mathbb{R} \setminus \{0\}$ (all real numbers except zero)

Key properties:

• The function is odd: $f(-x) = -f(x)$
• There is a horizontal asymptote with equation $y = 0$ (the x-axis)
• There is a vertical asymptote with equation $x = 0$ (the y-axis)

Functions with even negative integer powers

For $f(x) = x^n$ where $n$ is an even negative integer, we can write $f(x) = x^{-k}$ for $k = 2, 4, 6, \ldots$

The simplest case is $f(x) = x^{-2} = \frac{1}{x^2}$.

Domain and range:

• Maximal domain: $\mathbb{R} \setminus \{0\}$ (all real numbers except zero)
• Range: $\mathbb{R}^+$ (positive real numbers only)

Key properties:

• The function is even: $f(-x) = f(x)$
• There is a horizontal asymptote with equation $y = 0$ (the x-axis)
• There is a vertical asymptote with equation $x = 0$ (the y-axis)

Power functions with fractional index

When we have $f(x) = x^{\frac{1}{n}}$ where $n$ is a positive integer, we are dealing with root functions.

Definition of fractional powers

For a positive real number $a$ and natural number $n$, the expression $a^{\frac{1}{n}}$ represents the $n$th root of $a$. This is the positive number whose $n$th power equals $a$.

We can write this in two equivalent ways:

$$a^{\frac{1}{n}} = \sqrt[n]{a} \text{ where } (a^{\frac{1}{n}})^n = a$$

Examples:

• $9^{\frac{1}{2}} = 3$ because $3^2 = 9$
• $8^{\frac{1}{3}} = 2$ because $2^3 = 8$

Negative bases: When $n$ is odd, we can also define $a^{\frac{1}{n}}$ for negative values of $a$. In this case, $a^{\frac{1}{n}}$ is the number whose $n$th power equals $a$.

For example: $(-8)^{\frac{1}{3}} = -2$ because $(-2)^3 = -8$

Graphs of root functions

The domain of $f(x) = x^{\frac{1}{n}}$ depends on whether $n$ is even or odd:

• When $n$ is even: maximal domain is $\mathbb{R}^+ \cup \{0\}$ (non-negative numbers only)
• When $n$ is odd: maximal domain is $\mathbb{R}$ (all real numbers)

Inverses of power functions

Power functions with odd positive integer exponents are strictly increasing across all real numbers, which makes them one-to-one functions. This means they have inverse functions.

Important property

An important mathematical result states:

The proof of this involves showing that for any two values $a > b$, we always have $f(a) > f(b)$ by considering various cases based on whether the values are positive, negative, or zero.

Inverse of odd functions

Another key property relates to odd functions:

This can be proved by showing that if $y = f^{-1}(x)$, then from the odd property of $f$, we get $f(-y) = -x$, which implies $f^{-1}(-x) = -y = -f^{-1}(x)$.

Because of this property, when $n$ is odd, the function $f(x) = x^{\frac{1}{n}}$ is an odd function.

Remember!