Root functions

With an odd root function (n = 1,3,5, ...)

is defined by all real numbers x

With an even root function (n = 2,4,6, ...)

Is defined only by positive real numbers x. For example

is not defined.

Example 1

What is the domain of the function f(x)?

An even root function is defined when the root is non-negative.

Example 2

Solve the equation

Both sides of the equation are positive and in addition to this, the root is positive for all values of the variable x. The equation can be raised to an even power without additional conditions.

Example 3

solve the equation

Both sides of the equation are positive. The equation can be raised to an even power by considering the domain of the definition of the equation. The equation is defined when

We solve the equation

The answer fulfills the domain of the definition of the equation, so it is valid as a solution of the equation.

Example 4

Solve the equation

The equation can be raised to odd power without additional conditions.

Example 5

Solve the equation

Domain of definition

When the equation is raised to a second power, both sides of the equation must be positive (or zero). Let's form a so-called squaring definition:

We solve the equation taking these conditions into account.

We take all terms to the left of the equation to get a quadratic equation

Solutions

Both roots satisfy the equation definition but only x = 4 satisfies the squaring definition. The solution of the equation is

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