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In mathematics, root functions involve taking the root of a variable expression. Common examples include square roots, cube roots, and higher-order roots, such as the sixth root in this case. For example, the function \[ h(z) = -\sqrt[6]{5z + 2} \] is a sixth root function.

Root functions can look complex, but they are simply asking: 'What number, when raised to a certain power, results in the radicand?' Here, the radicand is the expression \( 5z + 2 \). Since we are dealing with a sixth root, we are looking for the sixth root of the radicand.

An important note: for even-order roots (like square roots, fourth roots, etc.), the radicand must be greater than or equal to zero to ensure the result is a real number. This helps us determine the domain of the function.

Determining the domain of functions often requires solving inequalities. Inequalities allow us to find the set of input values that will produce real and defined outputs.

Let's use the inequality approach to find the domain of our function \[ -\sqrt[6]{5z + 2} \].

Since we are dealing with a sixth root, we need the radicand (\( 5z + 2 \)) to be non-negative for the function to be real. This can be expressed as: \[ 5z + 2 \geq 0 \] Now, we need to solve this inequality to find the values of \( z \) that satisfy it.

Subtract 2 from both sides: \[ 5z \geq -2 \] Then, divide both sides by 5: \[ z \geq -\frac{2}{5} \] Therefore, the function is real and defined for all values of \( z \) that are greater than or equal to \( -\frac{2}{5} \).

Domain restrictions are rules that limit the set of allowable inputs for a function. These restrictions ensure that the function produces real and defined outputs.

For root functions, even-order roots (e.g., square roots, fourth roots, etc.) must have non-negative radicands. For our function \[ h(z) = -\sqrt[6]{5z + 2} \] the domain restriction is given by the inequality: \[ 5z + 2 \geq 0 \] This inequality tells us that \( z \) must be greater than or equal to \( -\frac{2}{5} \).

Hence, the domain of the function \( h(z) \) is: \[ \{ z \mid z \geq -\frac{2}{5} \} \] Or in interval notation, \[ [ -\frac{2}{5}, \infty ) \] Meaning, all values of \( z \) from \( -\frac{2}{5} \) upwards, inclusive of \( -\frac{2}{5} \), are acceptable inputs for the function.