Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.5 Inverse Functions
Forms
A Radical Function is a function similar to a polynomial or rational, but with fractional exponents. For example, 
is a radical function. Sometimes these can also be called Root Functions because 
. In General, a radical function, r(x), can be represented as the sum of terms whose variables have exponents that are part of the rational numbers.
Graph
Again, similar to polynomials, the exponent can determine a lot of information about the function. For some function 
where a/b is in simplified form, there are three cases:
1. If a and b are both odd, then the graph takes place in quadrants 1 and 3.
2. If a is odd b is even, then the graph only takes place in quadrant 1.
3. If a is even and b is odd, then the graph takes place in quadrants 2 and 3.
If a and b are both even, then a/b can be simplified! It will be one of the previous cases.
Additionally, if the top exponent is larger than the bottom exponent, then the graphs will curve upwards while each of the pictures below depict the opposite case.
Example 1: Graph y=x3/2+5
Solution:
Based on the exponent, we know that it mataches the second case, with the top exponent being larger. We only need to translate it up five units and switch the direction of the curve.
Example 2: Graph 
Solution:
Using a strategy similar to the first example, we rewrite the equation as 4-2x3/5 then apply the necessary changes.
Domain
We will refer to the three cases listed above.
1. If a and b are both odd, then the domain is unrestricted.
2. If a is odd b is even, then the domain is only positive numbers and zero. That is to say, x∈[0,∞).
3. If a is even and b is odd, then the domain is unrestricted.
Range
Again, we will refer to the three cases listed above.
1. If a and b are both odd, then the range is unrestricted.
2. If a is odd b is even, then the range is only positive numbers and zero. That is to say, y∈[0,∞).
3. If a is even and b is odd, then the range is again only positive numbers and zero. That is to say, y∈[0,∞).
Example 4: Identify the domain and range of 
Solution: It might help to rewrite this in exponential form: y=3(x-2)3/2+1
In this form it is clear that is matches the x(odd/even) format, so it has domain where the radicand is greater than or equal to 0: x-2≥0 or x≥2.
Additionally, the range of (x-2)(3/2) is y≥0 so f(x)=3(x-2)(3/2)+1 has a range of y≥1.
Special Forms
A generalized form of a radical function that demonstrates all of the possible transformations could appear as:
c represents the vertical stretch as well as whether or not the graph is reflected across the x-axis.
d represented the horizontal stretching factor as well as the reflection across the y-axis.
a and b represent the exponent.
h is the horizontal translation and k is the vertical translation.
Example 5: Graph 
Solution: This graph matches for form of a odd and b even, so we can use that shape with a few modifications:
c=2 tells us it is stretched vertically by a factor of 2
(1/3 x-4) factors to 1/3(x-12) so it is shifted horizontally to the right by 12 units
k=-2 means that the function is shifted down 2 units
Review
A Radical Function is a function with fractional exponents.
The Domain of a Radical Function depends on the exponent.
The Range of a Radical Function depends on the exponent.