## Fraction Power Function

A fraction power function is when our power function is to the power of a fraction rather than a whole number. There are two types of fraction positive and negative.

### Power Function with a positive fraction

$f(x)=x^{\frac{1}{n}}=\sqrt[n]{x}$

Where n is any positive integer.

An important concept that we need to remember for these functions are the difference between even and odd functions.

#### When n is even

When n is even, our function has no values along the negative x axis since we cannot have the squared root of a negative number.

Domain: $[0,\infty^{+})$

#### When n is odd

When n is odd our function has the domain of all real numbers, as we can have the cubed root of a negative number, as well as a positive number.

Domain: $(\infty^{-},\infty^{+})$

### Positive Fraction Power Function example

Take note of the different domains for when n is odd and when it is even.

These basic functions are not too hard to draw.

Step 1: Identify whether they are the square root of an even or odd number to determine the domain.

• We can see that the square root of x is to an even number. Therefore it is only in the first quadrant of the graph.
• We can see that the cubed root of x has an odd value. Therefore it will be in the first and third quadrants.

Step 2: Find the intercepts.

When n is even:

$f(0)=\sqrt{0}=0$

The function starts at the origin.

$f(1)=\sqrt{1}=1$

When n is even:

$f(0)=\sqrt[3]{0}=0$

The function goes through the origin.

$f(1)=\sqrt[3]{1}=1$

$f(-1)=\sqrt[3]{-1}=-1$

Once we find these numbers and remember the basic shape the function is easy to draw.

Remember it is also a good idea when drawing these graphs by hand to label another random point so that we know we have drawn the graph correctly and the examiner can see that it is the correct graph as well!

## Power Function with a Negative Fraction

$f(x)=x^{-\frac{1}{n}}=\frac{1}{\sqrt[n]{x}}$

### Sketching Power Function with a Negative Fraction

Step 1:  Identify the domain.

If n is even these functions the domain will be 0 not included to infinity.

Domain: $(0,\infty^{+})$

If n is odd all real values bar 0.

Domain: $(\infty^{-},\infty^{+})$

The main difference here between the positive power fractions is that the domain does not include zero since we can never have a number divide by zero.

Step 2: Identify the asymptotes.

The function is undefined when x equals 0, however will increase exponentially when it approaches 0. Therefore the first assymptote is x=0.

As x approaches infinity,the fraction will grow increasingly smaller, however it will never equal 0. Therefore tthe second assymptote y=0.

Step 3: Identify other key points

When n is even:

$f(1)=\frac{1}{sqrt{1}}=1$

When n is even:

$f(1)=\frac{1}{\sqrt[3]{1}}=1$

$f(-1)=\frac{1}{\sqrt[3]{-1}}=-1$

Step 4: Draw the graph to the correct shape

To finish drawing the graph the correct shape needs to be known. It is also a good idea to add in the co-ordinates of another point to ensure that you are correct and to show the marker you are correct.

### Negative Fraction Power Function example

$y=\frac{1}{\sqrt[3]{x}}$

First we can see it is a negative power fraction function with a n value of three therefore  the Domain is R\{0}

Second as there is no transformation it will go through the points (-1,-1) and (1,1).

Thirdly it will have asymptotes at x=0 and y=0.

Fourthly follow in the general shape of the graph moving from the points (-1,-1) and (1,1) to the two asymptotes and include another point!