More Power Functions Revision Notes for VCE SSCE Mathematical Methods

More Power Functions

Introduction

Previously, we have studied power functions of the form $f(x) = x^n$ , $f(x) = x^{-n}$ , and $f(x) = x^{\frac{1}{n}}$ , where $n$ is a positive integer. We also explored how to transform these functions. Now we will examine additional types of power functions that complete our understanding of this family of functions.

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In this section, we'll explore two important types of power functions: those with negative fractional powers ( $x^{-1/n}$ ) and those with general fractional powers ( $x^{p/q}$ ). Understanding these functions will complete your knowledge of power functions and their behaviors.

The function $f(x) = x^{-\frac{1}{n}}$ where $n$ is a positive integer

Definition and notation

A function with a negative fractional power can be written in several equivalent ways. The function $f(x) = x^{-\frac{1}{n}}$ can be expressed as:

x^{-\frac{1}{n}} = \frac{1}{x^{\frac{1}{n}}} = \frac{1}{\sqrt[n]{x}}

This shows us that a negative fractional power creates a reciprocal of the corresponding root function.

Domain considerations

The maximal domain of the function $f(x) = \frac{1}{\sqrt[n]{x}}$ depends on whether $n$ is odd or even:

When $n$ is odd: the maximal domain is $\mathbb{R} \setminus \{0\}$ (all real numbers except zero)
When $n$ is even: the maximal domain is $\mathbb{R}^+$ (positive real numbers only)

chatImportant

The domain excludes zero in both cases because division by zero is undefined. For even values of $n$ , we also exclude negative numbers because even roots of negative numbers are not real.

Quick rule: "Odd $n$ lets negatives in, even $n$ needs positive values"

Graph characteristics

Let's examine how these functions behave graphically by comparing them to the basic reciprocal function $y = \frac{1}{x}$ .

The graphs show important features common to all functions of the form $f(x) = x^{-\frac{1}{n}}$ :

Each graph has a horizontal asymptote with equation $y = 0$ (the function approaches zero as $x$ increases)
Each graph has a vertical asymptote with equation $x = 0$ (the function becomes unbounded near zero)
All graphs pass through the point $(1, 1)$
When $n$ is odd, the graph exists in both positive and negative regions and also passes through $(-1, -1)$

Odd and even function properties

An important property emerges when $n$ is an odd positive integer: the function $f(x) = x^{-\frac{1}{n}}$ is an odd function. This means:

f(-x) = -f(x)

This symmetry property means the graph is symmetric about the origin (rotational symmetry of 180°).

Worked example

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Worked Example: Sketching Graphs of Functions with Negative Fractional Powers

Question: For each of the following, sketch the graph of $y = f(x)$ for the maximal domain. State the maximal domain, the range, and whether the function is odd, even, or neither.

a) $f(x) = 3x^{-\frac{1}{4}} - 1$

b) $f(x) = 6x^{-\frac{1}{5}} + 1$

Solution:

Part a: $f(x) = 3x^{-\frac{1}{4}} - 1$

The maximal domain is $\mathbb{R}^+$ (because the power has an even denominator).

The range is $(-1, \infty)$ .

The function is neither odd nor even.

To find the $x$ -axis intercept, we set $y = 0$ :

3x^{-\frac{1}{4}} - 1 = 0

x^{-\frac{1}{4}} = \frac{1}{3}

x^{\frac{1}{4}} = 3

\therefore x = 3^4 = 81

The graph shows a curve with a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = -1$ .

Part b: $f(x) = 6x^{-\frac{1}{5}} + 1$

The maximal domain is $\mathbb{R} \setminus \{0\}$ (because the power has an odd denominator).

The range is $\mathbb{R} \setminus \{1\}$ (all real numbers except 1).

The function is neither odd nor even.

The line $x = 0$ is a vertical asymptote.

The line $y = 1$ is a horizontal asymptote.

The function $f(x) = x^{\frac{p}{q}}$ where $p$ and $q$ are positive integers

Definition and evaluation

When we have a fractional power where both the numerator and denominator are positive integers, we can express it in two equivalent ways:

x^{\frac{p}{q}} = (x^{\frac{1}{q}})^p = (\sqrt[q]{x})^p

chatImportant

Critical Rule: To use this definition correctly, we must always first write the fractional power in simplest form. For example, $x^{\frac{4}{6}}$ should be simplified to $x^{\frac{2}{3}}$ before evaluation.

Domain considerations

For the function $f(x) = x^{\frac{p}{q}}$ (where $\frac{p}{q}$ is in simplest form), the maximal domain depends on whether $q$ is odd or even:

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

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The denominator $q$ determines what type of root we're taking, and even roots require non-negative inputs. This is why $q$ (not $p$ ) determines the domain restrictions.

Evaluating fractional powers

Here are some examples showing how to evaluate expressions with fractional powers:

8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4

(-8)^{\frac{2}{3}} = ((-8)^{\frac{1}{3}})^2 = (-2)^2 = 4

10\,000^{\frac{3}{4}} = (10\,000^{\frac{1}{4}})^3 = 10^3 = 1000

0.0001^{\frac{3}{4}} = (0.0001^{\frac{1}{4}})^3 = 0.1^3 = 0.001

Notice that even when the base is negative (like $-8$ ), if the power in simplest form has an even numerator $p$ , the result will be positive.

Worked example

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Worked Example: Sketching Graphs of Functions with General Fractional Powers

Question: For each of the following, sketch the graph of $y = f(x)$ for the maximal domain. State the maximal domain, the range, and whether the function is odd, even, or neither.

a) $f(x) = x^{\frac{2}{3}}$

b) $f(x) = x^{\frac{3}{2}}$

Solution:

Part a: $f(x) = x^{\frac{2}{3}}$

The maximal domain is $\mathbb{R}$ (because the denominator 3 is odd).

The range is $[0, \infty)$ (all non-negative real numbers).

The function $f(x) = x^{\frac{2}{3}}$ is even, since $f(-x) = f(x)$ .

The graph has a V-shape opening upward from the origin, showing symmetry about the $y$ -axis (characteristic of even functions).

Part b: $f(x) = x^{\frac{3}{2}}$

The maximal domain is $[0, \infty)$ (because the denominator 2 is even).

The range is $[0, \infty)$ .

The function $f(x) = x^{\frac{3}{2}}$ is neither odd nor even.

The graph starts at the origin and curves upward, existing only for non-negative $x$ values.

Remember!

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Key Points to Remember:

For $f(x) = x^{-\frac{1}{n}}$ : The domain is $\mathbb{R} \setminus \{0\}$ if $n$ is odd, or $\mathbb{R}^+$ if $n$ is even. These functions always have asymptotes at $x = 0$ and $y = 0$ .
When $n$ is odd in $f(x) = x^{-\frac{1}{n}}$ , the function is odd (symmetric about the origin).
For $f(x) = x^{\frac{p}{q}}$ (in simplest form): The domain is $\mathbb{R}$ if $q$ is odd, or $\mathbb{R}^+ \cup \{0\}$ if $q$ is even.
Always simplify fractional powers before evaluating: $x^{\frac{p}{q}} = (\sqrt[q]{x})^p$
The denominator $q$ determines the domain (whether negative inputs are allowed), while the numerator $p$ affects the shape of the graph.

More Power Functions (VCE SSCE Mathematical Methods): Revision Notes