Roots and Radicals
Use Radicals in Functions
Learning Objectives
By the end of this section, you will be able to:
- Evaluate a radical function
- Find the domain of a radical function
- Graph radical functions
Before you get started, take this readiness quiz.
Evaluate a Radical Function
In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a radical function.
The square root function is
The cube root function is
A radical function is a function that is defined by a radical expression.
To evaluate a radical function, we find the value of f(x) for a given value of x just as we did in our previous work with functions.
For the function find ⓐ ⓑ
ⓐ
ⓑ
Since the square root of a negative number is not a real number, the function does not have a value at
For the function find ⓐ ⓑ
ⓐⓑ no value at
For the function find ⓐ ⓑ
ⓐⓑ no value at
We follow the same procedure to evaluate cube roots.
For the function find ⓐ ⓑ
ⓐ
ⓑ
For the function find ⓐ ⓑ
ⓐⓑ
For the function find ⓐ ⓑ
ⓐ
ⓑ
The next example has fourth roots.
For the function find ⓐ ⓑ
ⓐ
ⓑ
Since the fourth root of a negative number is not a real number, the function does not have a value at
For the function find ⓐ ⓑ
ⓐⓑ
For the function find ⓐ ⓑ
ⓐⓑ
Find the Domain of a Radical Function
To find the domain and range of radical functions, we use our properties of radicals. For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. For an odd index, the radicand can be any real number. We restate the properties here for reference.
When n is an even number and:
- then is a real number.
- then is not a real number.
When n is an odd number, is a real number for all values of a.
So, to find the domain of a radical function with even index, we set the radicand to be greater than or equal to zero. For an odd index radical, the radicand can be any real number.
When the index of the radical is even, the radicand must be greater than or equal to zero.
When the index of the radical is odd, the radicand can be any real number.
Find the domain of the function, Write the domain in interval notation.
Since the function, has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0. We set the radicand to be greater than or equal to 0 and then solve to find the domain.
The domain of is all values and we write it in interval notation as
Find the domain of the function, Write the domain in interval notation.
Find the domain of the function, Write the domain in interval notation.
Find the domain of the function, Write the domain in interval notation.
Since the function, has a radical with an index of 2, which is even, we know the radicand must be greater than or equal to 0.
The radicand cannot be zero since the numerator is not zero.
For to be greater than zero, the denominator must be positive since the nu