Power functions with negative exponents

Functions: Fractional functions

Power functions with negative exponents

Power function with negative exponent

A power function with a negative integer exponent has the form $f(x)=\blue{a}x^{-\orange{n}}$

in which $\orange{n}$ is a postive integer.

We can also write this function as $f(x)=\frac{\blue{a}}{x^{\orange{n}}}$

The graph of a power function with a negative integer exponent, moves through the point $\rv{1,\blue{a}}$ , has a vertical asymptote at $x=0$ and horizontal asymptote in the line $y=0$ .

If $\orange{n}$ is even, the function is symmetrical across the $y$ -axis. If $\orange{n}$ is odd, the function has the point $\rv{0,0}$ as the point of symmetry.

GeoGebra Negative powerfunction

A power function with negative exponent has a vertical asymptote at $x=0$ . This is so because we are not allowed to divide by $0$ , hence, the function value at $f(0)$ does not exist. The domain of a power function with negative exponent are all numbers, except $x=0$ .

The range of a power function with an even negative exponent is equal to all positive numbers if $\blue a \gt 0$ . If $\blue a \lt 0$ , then the range is equal to all negative numbers.

For a power function with odd negative exponent, the range is equal to all numbers except $0$ .

Ev‍en and odd function

When $\orange{n}$ is even, we call $f(x)=\frac{\blue{a}}{x^{\orange{n}}}$ an even function. This means that: $f(-x)=f(x)$

Hence, this is another way of describing that the graph is symmetrical across the $y$ -axis.

When $\orange{n}$ is odd, we call $g(x)=\frac{\blue{a}}{x^{\orange{n}}}$ an odd function. This means that: $g(-x)=-g(x)$

Hence, this is another way of describing that the graph has point symmetry across the point $\rv{0,0}$ .

Example

For $f(x)=\frac{\blue{1}}{x^{\orange{4}}}$ we have:

$f(-2)=\frac{\blue1}{\left(-2\right)^{\orange{4}}}=\frac{1}{16}=\frac{\blue1}{\left(2\right)^{\orange{4}}}=f(2)$

For $g(x)=\frac{\blue1}{x^{\orange{3}}}$ we have:

$f(-2)=\frac{\blue1}{\left(-2\right)^{\orange{3}}}=\frac{1}{8}=-\frac{\blue1}{\left(2\right)^{\orange{3}}}=-f(2)$

Take a look at the graph of a power function with negative exponent, which is a function of the form $f(x)=\frac{a}{x^n}$ .

What do we know about the values of $n$ and $a$ ?

The value of $n$ is: odd
The value of $a$ is: positive

The graph is symmetrical across the point $\rv{0,0}$ , hence, the value of $n$ is odd.
The $y$ -value is positive if the value of $x$ is positive, hence the value of $a$ is positive.

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