AOS1 Topic 5: Power Functions

A power function is a type of mathematical function that can be represented in the form:

\(f(x) = k \cdot x^n\)

where:

\(k\) is a constant (called the coefficient),

\(n\) is a real number (called the exponent),

\(x\) is the variable.

Power functions are a specific type of polynomial function. The behavior and shape of the graph of a power function depend on the values of \(k\) and \(n\).

Key Characteristics of Power Functions Based on Different Values of

\(n\):

Integer Exponents:

If \(n\) is a positive integer, the function is a polynomial of degree \(n\).

For example \(f(x) = x^2\) is a quadratic function, and \(f(x) = x^3\) is a cubic function.

If \(n = 1\), the function is a linear function: \(f(x) = kx\).

If \(n = 0\), the function is a constant function: \(f(x) = k\).

Negative Exponents:

If \(n\) is a negative integer, the function represents a reciprocal function.

For example, \(f(x) = x^{-1}\) or \(f(x) = \frac{1}{x}\).

Fractional Exponents:

If \(n\) is a fraction, the function represents a root function.

For example, \(f(x) = x^{\frac{1}{2}}\) or \(f(x) = \sqrt{x}\).

Even and Odd Functions:

If \(n\) is even, the graph of the function is symmetric with respect to the y-axis (even function).

For example,/strong>, \(f(x) = x^2\) is symmetric about the y-axis.

If \(n\) is odd, the graph of the function is symmetric with respect to the origin (odd function).

For example, \(f(x) = x^3\) is symmetric about the origin.

Examples:

Quadratic Function (a specific type of power function where \(n = 2\)):

\(f(x) = 3x^2\)

The graph is a parabola opening upwards with vertex at the origin.

Cubic Function (where \(n = 3\)):

\(f(x) = 2x^3\)

The graph has an inflection point at the origin and increases more steeply compared to a linear function.

Reciprocal Function (where \(n = -1\)):

\(f(x) = \frac{1}{x}\)

The graph has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Square Root Function (where \(n = \frac{1}{2}\)):

\(f(x) = \sqrt{x}\)

The graph starts at the origin and increases slowly, remaining in the first quadrant.

Increasing and Decreasing Functions

Strictly Increasing Functions

We say a function \( f \) is strictly increasing on an interval if \( x_2 > x_1 \) implies \( f(x_2) > f(x_1) \).

For example:

  • The graph below shows a strictly increasing function.
  • A straight line with positive gradient is strictly increasing.
  • The function \( f : [0, \infty) \to \mathbb{R}, \, f(x) = x^2 \) is strictly increasing.

Strictly Decreasing Functions

We say a function \( f \) is strictly decreasing on an interval if \( x_2 > x_1 \) implies \( f(x_2) < f(x_1) \).

For example:

  • The graph below shows a strictly decreasing function.
  • A straight line with negative gradient is strictly decreasing.
  • The function \( f : (-\infty, 0] \to \mathbb{R}, \, f(x) = x^2 \) is strictly decreasing.

Power Functions with Positive Integer Index

We start by considering power functions \( f(x) = x^n \) where \( n \) is a positive integer.

Taking \( n = 1, 2, 3 \), we obtain the linear function \( f(x) = x \), the quadratic function \( f(x) = x^2 \), and the cubic function \( f(x) = x^3 \).

The general shape of the graph of \( f(x) = x^n \) depends on whether the index \( n \) is odd or even.

The Function \( f(x) = x^n \) where \( n \) is an Odd Positive Integer

The graph has a similar shape to those shown below. The maximal domain is \( \mathbb{R} \) and the range is \( \mathbb{R} \).

Some properties of \( f(x) = x^n \) where \( n \) is an odd positive integer:

  • \( f \) is an odd function
  • \( f \) is strictly increasing
  • \( f \) is one-to-one
  • \( f(0) = 0 \), \( f(1) = 1 \), and \( f(-1) = -1 \)
  • As \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \)

The Function \( f(x) = x^n \) where \( n \) is an Even Positive Integer

The graph has a similar shape to those shown below. The maximal domain is \( \mathbb{R} \) and the range is \( \mathbb{R}^+ \cup \{0\} \).

Some properties of \( f(x) = x^n \) where \( n \) is an even positive integer:

  • \( f \) is an even function
  • \( f \) is strictly increasing for \( x \geq 0 \)
  • \( f \) is strictly decreasing for \( x \leq 0 \)
  • \( f(0) = 0 \), \( f(1) = 1 \), and \( f(-1) = 1 \)
  • As \( x \to \pm\infty \), \( f(x) \to \infty \)

Note: The function \( f \) is strictly increasing for \( x \in [0, \infty) \) and strictly decreasing for \( (-\infty, 0] \).

Power Functions with Negative Integer Index

Again, the general shape of the graph depends on whether the index \( n \) is odd or even.

The function \( f(x) = x^n \) where \( n \) is an odd negative integer

Taking \( n = -1 \), we obtain

\( f(x) = x^{-1} = \frac{1}{x} \)

The graph of this function is shown on the right. The graphs of functions of this type are all similar to this one.

In general, we consider the functions \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R}, f(x) = x^{-k} \) for \( k = 1, 3, 5, \ldots \)

  • The maximal domain is \( \mathbb{R} \setminus \{0\} \) and the range is \( \mathbb{R} \setminus \{0\} \).
  • \( f \) is an odd function.
  • There is a horizontal asymptote with equation \( y = 0 \).
  • There is a vertical asymptote with equation \( x = 0 \).

    • The Function \( f(x) = x^{1/n} \) Where \( n \) is a Positive Integer

    Let \( a \) be a positive real number and let \( n \in \mathbb{N} \). Then \( a^{1/n} \) is defined to be the nth root of \( a \). That is, \( a^{1/n} \) is the positive number whose nth power is \( a \). We can also write this as \( a^{1/n} = \sqrt[n]{a} \).

    For example: \( 9^{1/2} = 3 \), since \( 3^2 = 9 \).

    We define \( 0^{1/n} = 0 \), for each natural number \( n \), since \( 0^n = 0 \).

    If \( n \) is odd, then we can also define \( a^{1/n} \) when \( a \) is negative. If \( a \) is negative and \( n \) is odd, define \( a^{1/n} \) to be the number whose nth power is \( a \). For example: \( (-8)^{1/3} = -2 \), as \( (-2)^3 = -8 \).

    In all three cases we can write:

    \[ a^{1/n} = \sqrt[n]{a} \text{ with } \left( a^{1/n} \right)^n = a \]

    In particular, \( x^{1/2} = \sqrt{x} \).

    Let \( f(x) = x^{1/n} \). When \( n \) is even, the maximal domain is \( \mathbb{R}^{+} \cup \{0\} \) and when \( n \) is odd, the maximal domain is \( \mathbb{R} \). The graphs of \( f(x) = \sqrt{x} = x^{1/2} \) and \( f(x) = \sqrt[3]{x} = x^{1/3} \) are as shown.

    Inverses of Power Functions

    Proof for Odd Positive Integer \(n\)

    We prove the following result in the special case when \( n = 5 \). The general proof is similar.

    If \( n \) is an odd positive integer, then \( f(x) = x^n \) is strictly increasing for \( \mathbb{R} \).

    Proof

    Let \( f(x) = x^5 \) and let \( a > b \). To show that \( f(a) > f(b) \), we consider five cases.

    Case 1: \( a > b > 0 \)

    We have

    \( f(a) - f(b) = a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) \) (Show by expanding.)

    Since \( a > b \), we have \( a - b > 0 \). Since we are assuming that \( a \) and \( b \) are positive in this case, all the terms of \( a^4 + a^3b + a^2b^2 + ab^3 + b^4 \) are positive. Therefore \( f(a) - f(b) > 0 \) and so \( f(a) > f(b) \).

    Case 2: \( a > 0 \) and \( b < 0 \)

    In this case, we have \( f(a) = a^5 > 0 \) and \( f(b) = b^5 < 0 \) (an odd power of a negative number). Thus \( f(a) > f(b) \).

    Case 3: \( a = 0 \) and \( b < 0 \)

    We have \( f(a) = 0 \) and \( f(b) < 0 \). Thus \( f(a) > f(b) \).

    Case 4: \( b = 0 \) and \( a > 0 \)

    We have \( f(a) > 0 \) and \( f(b) = 0 \). Thus \( f(a) > f(b) \).

    Case 5: \( 0 > a > b \)

    Let \( a = -c \) and \( b = -d \), where \( c \) and \( d \) are positive. Then \( a > b \) implies \( -c > -d \) and so \( c < d \). Hence \( f(c) < f(d) \) by Case 1 and thus \( f(-a) < f(-b) \). But \( f \) is an odd function and so \( -f(a) < -f(b) \). Finally, we have \( f(a) > f(b) \).

    Note: For the general proof, use the identity

    \( a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \cdots + a^2b^{n-3} + ab^{n-2} + b^{n-1}) \)

    If \( f \) is a strictly increasing function on \( \mathbb{R} \), then it is a one-to-one function and so has an inverse. Thus \( f(x) = x^n \) has an inverse function, where \( n \) is an odd positive integer.

    Similar results can be achieved for restrictions of functions with rules \( f(x) = x^n \), where \( n \) is an even positive integer. For example, \( g: \mathbb{R}^+ \cup \{0\} \to \mathbb{R} \), \( g(x) = x^6 \) is a strictly increasing function and \( h: \mathbb{R}^- \cup \{0\} \to \mathbb{R} \), \( h(x) = x^6 \) is a strictly decreasing function. In both cases, these restricted functions are one-to-one.

    Odd One-to-One Functions

    If \( f \) is an odd one-to-one function, then \( f^{-1} \) is also an odd function.

    Proof: Let \( x \in \text{dom } f^{-1} \) and let \( y = f^{-1}(x) \). Then \( f(y) = x \). Since \( f \) is an odd function, we have \( f(-y) = -x \), which implies that \( f^{-1}(-x) = -y \). Hence \( f^{-1}(-x) = -f^{-1}(x) \).

    By this result we see that, if \( n \) is odd, then \( f(x) = x^{\frac{1}{n}} \) is an odd function. It can also be shown that, if \( f \) is a strictly increasing function, then \( f^{-1} \) is strictly increasing.

    Example 1

    Function Analysis

    For the function \( f \) with rule \( f(x) = \frac{1}{x^5} \):

  • State the maximal domain and the corresponding range.
  • Evaluate each of the following:
    • i. \( f(2) \)
    • ii. \( f(-2) \)
    • iii. \( f\left(\frac{1}{2}\right) \)
    • iv. \( f\left(-\frac{1}{2}\right) \)
  • Sketch the graph without using your calculator.

    • Solution:a The maximal domain is \( \mathbb{R} \setminus \{0\} \) and the range is \( \mathbb{R} \setminus \{0\} \).

    b

    • i \( f(2) = \frac{1}{2^5} = \frac{1}{32} \)
    • ii \( f(-2) = \frac{1}{(-2)^5} = -\frac{1}{32} \)
    • iii \( f\left(\frac{1}{2}\right) = \frac{1}{\left(\frac{1}{2}\right)^5} = 32 \)
    • iv \( f\left(-\frac{1}{2}\right) = \frac{1}{\left(-\frac{1}{2}\right)^5} = -32 \)

    Graph Sketching:


    Example 2

    Function Comparison

    Let \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R} \), \( f(x) = x^{-1} \) and \( g : \mathbb{R} \setminus \{0\} \to \mathbb{R} \), \( g(x) = x^{-3} \).

    a Find the values of \( x \) for which \( f(x) = g(x) \).

    b Sketch the graphs of \( y = f(x) \) and \( y = g(x) \) on the one set of axes.

    Solution:

    To find the values of \( x \) for which \( f(x) = g(x) \), we set the equations equal to each other:

    \( f(x) = g(x) \)

    \( x^{-1} = x^{-3} \)

    By simplifying, we get:

    \( x^{-1} = x^{-3} \)

    \( x^2 = 1 \)

    \( x = \pm 1 \)

    So, the values of \( x \) for which \( f(x) = g(x) \) are \( x = 1 \) or \( x = -1 \).

    b.

    Graph sketching

    Note:

    • If \( x > 1 \), then \( x^3 > x \) and so \( \frac{1}{x} > \frac{1}{x^3} \).
    • If \( 0 < x < 1 \), then \( x^3 < x \) and so \( \frac{1}{x} < \frac{1}{x^3} \).
    • If \( x < -1 \), then \( x^3 < x \) and so \( \frac{1}{x} < \frac{1}{x^3} \).
    • If \( -1 < x < 0 \), then \( x^3 > x \) and so \( \frac{1}{x} > \frac{1}{x^3} \).

    Example 3

    Let \( f : \mathbb{R} \rightarrow \mathbb{R}, \, f(x) = x^{1/3} \) and \( g: \mathbb{R}^+ \cup \{0\} \rightarrow \mathbb{R}, \, g(x) = x^{1/2} \).

    Find the values of \( x \) for which \( f(x) = g(x) \).

    Sketch the graphs of \( y = f(x) \) and \( y = g(x) \) on the one set of axes.

    Solution:

    We need to solve the equation \( x^{1/3} = x^{1/2} \).

    Raising both sides to the power of 6 to eliminate the radicals, we get:

    \( (x^{1/3})^6 = (x^{1/2})^6 \)

    \( x^2 = x^3 \)

    Rearranging the equation, we get:

    \( x^3 - x^2 = 0 \)

    \( x^2 (x - 1) = 0 \)

    So, \( x = 0 \) or \( x = 1 \).

    Therefore, the values of \( x \) for which \( f(x) = g(x) \) are \( x = 0 \) and \( x = 1 \).

    Graph can be seen in the previous section


    Example 4

    Finding Inverse of the Function

    Find the inverse of the following function:

    \( f : (1, \infty) \to \mathbb{R}, \, f(x) = 64x^6 \)

    Solution:

    Function: \( f : (1, \infty) \to \mathbb{R}, \, f(x) = 64x^6 \)

    Note that \( f \) has range \( (64, \infty) \). Therefore \( f^{-1} \) has domain \( (64, \infty) \) and range \( (1, \infty) \).

    Write \( y = 64x^6 \). Interchange \( x \) and \( y \) and then solve for \( y \):

    \( x = 64y^6 \)

    \( y^6 = \frac{x}{64} \)

    \( \therefore y = \pm \left( \frac{x}{64} \right)^{\frac{1}{6}} \)

    Thus, \( f^{-1} : (64, \infty) \to \mathbb{R}, \, f^{-1}(x) = \frac{1}{2}x^{\frac{1}{6}} \)

    Example 5

    Find the inverse of each of the following functions:

    f : \( (-\infty, 0] \rightarrow \mathbb{R} \), \( f(x) = x^4 \)

    Solution:

    \( f : (-\infty, 0] \rightarrow \mathbb{R} \), \( f(x) = x^4 \)
    Note that \( f \) has range [0, ∞). Therefore \( f^{-1} \) has domain [0, ∞) and range (-∞, 0].
    Write \( y = x^4 \). Interchange \( x \) and \( y \) and then solve for \( y \):
    \( x = y^4 \)
    ∴ \( y = \pm x^{1/4} \)
    Thus \( f^{-1} \): [0, ∞) → \( \mathbb{R} \), \( f^{-1}(x) = -x^{1/4} \)

    Exercise &&1&& (&&1&& Question)

    What is the general form of a power function?

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    Exercise &&2&& (&&1&& Question)

    What is the behavior of the graph of a power function with an even positive integer index as \( x \) approaches infinity?

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    Exercise &&3&& (&&1&& Question)

    What happens to the graph of a power function with a positive integer index as \( x \) approaches negative infinity?

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    Exercise &&4&& (&&1&& Question)

    Find the Inverse of the following Function

    \( f : \mathbb{R} \to \mathbb{R}, \, f(x) = x^5 \)

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