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 \(n\) .



Created with GeoGebra  ®  , by Rabiya Manzoor, Link




Key Characteristics of Power Functions Based on Different Values of \(n\) :


1. 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


\(f(x) = x^3\) is a cubic function.


Special Cases:


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:


\(f(x) = x^{\frac{1}{2}} = \sqrt{x}\) .



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,


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


Red Graph: (\x^3\), Blue Graph: \(x^2\)






Examples of Power Functions:


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


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



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


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


Inflection Point: At the origin $(0,0)$.


Behavior: Increases more steeply compared to a linear function.


Symmetry: Odd function, symmetric about the origin.



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


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


Asymptotes:


Vertical Asymptote: $x = 0$.

Horizontal Asymptote: $y=0$.


Symmetry: Odd function, symmetric about the origin.


Behavior: Approaches infinity as $x$ approaches zero from the positive side and negative infinity from the negative side.


Note: An asymptote is where the function approaches the number but will never reach it.






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


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


Domain: $x \geq 0$.


Range: $y \geq 0$.


Shape: Starts at the origin and increases slowly, remaining in the first quadrant.






Increasing and Decreasing Functions


The function below is \(y = x(x- \frac{7}{4})(x+ \frac{7}{4})\) not $y=x^2$



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1. Strictly Increasing Functions


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


For example:


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 \( x_2 > x_1 \) and \( f(x_2) < f(x_1) \).


For example:


A straight line with negative gradient is strictly decreasing.


The function \( f : (-\infty, 0] \to \mathbb{R}, \, f(x) = x^2 \) is strictly decreasing.



Below is an example of both.

Created with GeoGebra  ®  , by katimars77, Link




Power Functions with Positive Integer Exponents


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


Taking \( n = 1, 2, 3 \), we get


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 .


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


The maximal domain is \( \mathbb{R} \)

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 graph has a similar shape to those shown below.





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


The maximal domain is \( \mathbb{R} \)

The range is \( \mathbb{R}^+ \cup \{0\} \). (If the coefficient in front is positive )


Note: \( \mathbb{R}^+\) means all real positive numbers \( \cup \{0\}\) means including 0


The range is \( \mathbb{R}^- \cup \{0\} \). (If the coefficient in front is negative )


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 \( x \leq 0 \)


Not One-to-One: Fails the horizontal line test, as multiple $x$ values can yield the same, \( f(0) = 0 \), \( f(1) = 1 \), 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 \( (-\infty, 0] \).


The graph has a similar shape to those shown below.





Power Functions with Negative Integer


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


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


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 \( \mathbb{R} \setminus \{0\} \)

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 \( f(x) = x^n \) where \( n \) is an even negative integer


Taking \( n = -2 \), we obtain


\( f(x) = x^{-2} = \frac{1}{x^2} \) ( Truncus )


In general, we consider the functions \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R}, f(x) = x^{-k} \) for \( k = 2, 4, 6, \ldots \)


The maximal domain is \( \mathbb{R} \setminus \{0\} \)

The range is \( \mathbb{R}^+ \setminus \{0\} \). (unless the coefficient is negative)


\( f \) is an even function.


There is a horizontal asymptote with equation \( y = 0 \).


There is a vertical asymptote with equation \( x = 0 \).


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


When $n$ is a fraction, the power function represents a root function.


We can also write this as \( x^{1/n} = \sqrt[n]{x} \).


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


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


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.


Created with GeoGebra ® , by Rabiya Manzoor, Link




Common Misconceptions


Misconception 1: Power Functions with Fractional Exponents are Always Defined for All Real Numbers


Clarification: For even fractional exponents (e.g., $x^{\frac{1}{2}}$​), the function is only defined for $x \geq 0$. For odd fractional exponents (e.g., $x^{\frac{1}{3}}​$), the function is defined for all real numbers.


Misconception 2: All Power Functions are Polynomials


Clarification: Only power functions with integer exponents are polynomials. Power functions with non-integer exponents (fractional or negative) are not polynomials.


Misconception 3: The Coefficient $k$ in Power Functions Affects Only the Vertical Stretch/Compression


Clarification: The coefficient $k$ affects both the vertical stretch/compression and the reflection over the x-axis if $k$ is negative.



Summary: Power Functions


Definition: A power function is a mathematical function of the form $f(x) = k \cdot x^n$, where $k$ is a constant, $n$ is a real number, and $x$ is the variable.


Key Characteristics Based on $n$:


Integer Exponents:


  1. Positive Integers: Polynomial functions (linear, quadratic, cubic, quartic).
  2. Negative Integers: Reciprocal functions.


Fractional Exponents: Root functions.


Even Exponents: Even functions, symmetric about the y-axis.


Odd Exponents: Odd functions, symmetric about the origin.


Function Types and Properties:

  1. Quadratic Function ($n = 2$): U-shaped parabola, even function.
  2. Cubic Function ($n = 3$): S-shaped curve, odd function.
  3. Reciprocal Function ($n = -1$): Hyperbola with asymptotes, odd function.
  4. Reciprocal Function ($n = -2$): Truncus with asymptotes, even function.


Root Function ($n = \frac{1}{2}$​): Starts at the origin, increases slowly, even function.


Increasing and Decreasing Behavior:


  1. Strictly Increasing: For odd exponents across $\mathbb{R}$; for even exponents on $[0, \infty)$.


  1. Strictly Decreasing: For even exponents on $(-\infty, 0]$.


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

    Exercise 2

    Exercise 3

    Exercise 4

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