Domain: The domain of every constant function is always \((-\infty , \infty )\).

Zeros: If the constant funciton just happens to be the zero function and every value is \(0\), then every number in the domain is a zero of the function.

Otherwise, constant functions do not have zeros.

Continuity: Constant functions are always continuous.

End-Behavior: The end-behavior is just the value of constant function.

Behavior (Inc/Dec): Constant functions do not increase and they do not decrease.

Global Maximum and Minimum: Since a constant function has only one value, that value is the global maxmum and global minimum and they both occur at every domain number.

Local Maximum and Minimum: Since a constant function has only one value, that value is a local maxmum and local minimum and they both occur at every domain number.

Range: The range of a constant function consists of just one number, the value of the function.

Domain: The domain of the identity function is \((-\infty , \infty )\).

Zeros: \(0\) is the only zero of the identity function.

Continuity: The identity function is a continuous function.

End-Behavior:

As the domain becomes unbounded positively, the identity also becomes unbounded positively. Our notation for this will look like

As the domain becomes unbounded negatively, the identity also becomes unbounded negatively. Our notation for this will look like

Behavior (Inc/Dec): The identity is an increasing function.

Global Maximum and Minimum: The identity function has no global maximum or global minimum. This is explained by the end-behavior.

Local Maximum and Minimum: The identity function has no local maximums or global minimums.

Range: The range of the identity function is \((-\infty , \infty )\).

The power makes a big difference here. As a beginning, we will only consider two types of powers: positive integers and negative integers,

Domain:

If \(n\) is a whole number, then the domain is \((-\infty , \infty )\).

If \(n\) is a negative integer, then the domain is \((-\infty , 0) \cup (0, \infty )\).

Zeros:

If \(n\) is a whole number, then \(0\) is the only zero.

If \(n\) is a negative integer, then the power function has no zeros.

Continuity: Power functions are continuous functions

End-Behavior:

If \(n\) is a whole number, then there are two different cases.

• If \(n\) is odd, then the power function is unbounded positively as the domain becomes unbounded positively or negative.
• If \(n\) is even, then the power function is unbounded positively in both directions.

If \(n\) is a negative integer, then the power function tends to \(0\) as the domain becomes unbounded positively or negatively.

Behavior (Inc/Dec):

If \(n\) is a whole number, then there are two different cases.

• If \(n\) is odd, then the power function is an increasing function.
• If \(n\) is even, then the power function decreases on \((-\infty , 0)\) and increases on \((0, \infty )\).

If \(n\) is a negative integer, then there are two different cases.

• If \(n\) is odd, then the power function is a decreasing function.
• If \(n\) is even, then the power function increases on \((-\infty , 0)\) and decreases on \((0, \infty )\).

Global Maximum and Minimum:

A power function has a global minimums only if the power is an even whole number. In that case, \(0\) is the global minimum, which occurs at \(0\).

Local Maximum and Minimum:

Range:

To begin, we are only considering \(n\) to be a natural number.

Domain:

• If \(n\) is odd, then the domain is \((-\infty , \infty )\).
• If \(n\) is even, then the domain is \([0. \infty )\).

Zeros: Core root functions only have \(0\) as their only zero.

Continuity: Core root functions are continuous functions.

End-Behavior:

• If \(n\) is odd, then the root function is unbounded positively as the domain becomes unbounded positively and unbounded negatively as the domain becomes unbounded negatively.

  Our notation for this will look like

  \[ \lim \limits _{x \to \infty } \sqrt [n]{x} = \infty \]
  \[ \lim \limits _{x \to -\infty } \sqrt [n]{x} = -\infty \]

• If \(n\) is even, then there is only end-behavior in the positive direction.

  \[ \lim \limits _{x \to \infty } \sqrt [n]{x} = \infty \]

Behavior (Inc/Dec): Core root functions are increasing functions.

Global Maximum and Minimum:

• If \(n\) is even, then the root function has \(0\) as its global minimum value and no maximum global value.
• If \(n\) is odd, then there is neither a global minimum nor global maximum.

Local Maximum and Minimum:

• If \(n\) is even, then the root function has \(0\) as its local minimum value and no local maximum values.
• If \(n\) is odd, then there are no local extrema.

Range:

• If \(n\) is even, then the range is \([0, \infty )\).
• If \(n\) is odd, then the range is \((-\infty , \infty )\).