10.1 Radical Expressions and Functions
A radical expression is any expression that can be written in the form
where n is the index (an integer ≥ 2), and a is the radicand. If the index is 2, it is usually not written. The symbol is called the radical sign.
In working with radical expressions, it is necessary to divide them into two groups: those with an even index and those with an odd index.
Expressions with an Even Index
- The radicand must be a nonnegative number (positive or zero).
It can’t be negative.The even root of a negative number is
“undefined” ora “nonreal number.”
[read “the square root of negative 3”] is a nonreal number
[read ‘the fourth root of negative 2”] is a nonreal number
- There are two roots:
The principal (positive) root is represented using .
The negative root is represented using .
If you want both roots, write and OR
because 72 = 49 because (–9)2 = 81
because 34 = 81 because (–2)4 = 16
Expressions with an Odd Index
- The radicand can be any real number (positive, negative or zero).
- There is only one root.
The root is positive if the radicand is positive.
The root is negative if the radicand is negative.
Roots are represented using .
because 43 = 64
because (–4)3 = –64
Radical Functions
A radical relation equates a radical expression in one variable to a second (dependent) variable.
If a radical relation is a radical function, it can be written using function notation.
A radical function is written in the form where is a radical expression with the variable x in the radicand.
not a radical function because x is not part of the radicand is a better way to write this
is a radical function
is a radical function
Be careful when identifying the radicand. It is only that part of the expression inside the radical sign.
x the radicand is 2 Better is
the radicand is 2x
the radicand is 2x+5
the radicand is 2x
Domain of a Radical Function
The domain of a function is the set of all real-number values for the independent variable that give a real–number result. Any value that
doesn’t give such a result is excluded from the domain and is referred to as a restricted value.
In radical expressions with an odd index, the radicand can be any real number. So, radical functions defined by odd roots have no restricted
values. The domain for these functions is all real numbers.
In radical expressions with an even index, the radicand must be positive or zero. These expressions are not defined for any negative number. So,
those real numbers that result in a negative radicand are restricted
valuesand must be excluded from the domain.
Finding the Domain Algebraically
- Set up an inequality of the form radicand
- Solve the inequality.
- Solutions to the inequality are the domain.
Example: Find the domain for
The radicand is 3x.
The domain is all real numbers greater than or equal to 0.
or
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Example: Find the domain for
The radicand is 5x – 4.
The domain is all real numbers greater than or equal to .
or
Example: Find the domain of the function
The radicand is 2x – 5.
The domain is all real numbers greater than or equal to
or
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Evaluating Radical Functions
To evaluate radical functions, substitute the given value for the variable and simplify. Do not leave the expression in radical form. Take the indicated root rounding to two decimal places if necessary.
Example: For
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Example: For
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Example: For
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Example: For
Simplifying Radical Expressions of the Form
Expressions with n even
This is the principal root and must be positive.
This is the negative root and must be negative.
Expressions with n odd
The sign of the root depends upon the sign of the radicand.