Fundamental Functions and Piecewise Functions in Precalculus

Fundamental Functions

Constant Function

The constant function is one of the simplest types of functions, where the output value is always the same regardless of the input.

• Definition: $f(x) = c$, where $c$ is a constant.

• Domain: $(-\infty, +\infty)$

• Range: Only the constant $c$

• Behavior: The function is constant on $(-\infty, +\infty)$ (neither increasing nor decreasing).

• Example: $f(x) = 3$

Identity Function

The identity function returns the input as the output. It is a fundamental linear function.

• Definition: $f(x) = x$

• Domain: $(-\infty, +\infty)$

• Range: $(-\infty, +\infty)$

• Behavior: Increasing on $(-\infty, +\infty)$

• Example: $f(2) = 2$

Absolute Value Function

The absolute value function outputs the non-negative value of its input, creating a 'V' shape on the graph.

• Definition: $f(x) = |x|$

• Domain: $(-\infty, +\infty)$

• Range: $[0, +\infty)$

• Behavior: Decreasing on $(-\infty, 0)$ and increasing on $(0, +\infty)$

• Example: $f(-3) = 3$

Square Root Function

The square root function is defined only for non-negative inputs and produces non-negative outputs.

• Definition: $f(x) = \sqrt{x}$

• Domain: $[0, +\infty)$

• Range: $[0, +\infty)$

• Behavior: Increasing on $[0, +\infty)$

• Example: $f(9) = 3$

Cubic Function

The cubic function is a polynomial of degree three, with both positive and negative outputs.

• Definition: $f(x) = x^3$

• Domain: $(-\infty, +\infty)$

• Range: $(-\infty, +\infty)$

• Behavior: Increasing on $(-\infty, +\infty)$

• Example: $f(-2) = -8$

Cubic Root Function

The cubic root function is defined for all real numbers and returns the real root of the input.

• Definition: $f(x) = \sqrt[3]{x}$

• Domain: $(-\infty, +\infty)$

• Range: $(-\infty, +\infty)$

• Behavior: Increasing on $(-\infty, +\infty)$

• Example: $f(-8) = -2$

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions over different intervals of the domain. Each 'piece' applies to a specific part of the domain.

• General Form: $f(x) = \begin{cases} \text{expression 1} & \text{if } x \text{ in interval 1} \\ \text{expression 2} & \text{if } x \text{ in interval 2} \end{cases}$

• Key Steps to Graph:

  1. Identify the intervals and corresponding expressions.

  2. Plot each piece on its specified interval.

  3. Check for continuity or jumps at the boundaries.

Example 1

Consider the function:

$f(x) = \begin{cases} 2x & \text{if } x \leq 1 \\ 2 & \text{if } x > 1 \end{cases}$

• For $x \leq 1$, plot the line $y = 2x$.

• For $x > 1$, plot the constant function $y = 2$.

• At $x = 1$, the value is $f(1) = 2$ (from the first piece).

Example 2

Consider the function:

$f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x-2 & \text{if } x < 0 \end{cases}$

• For $x \geq 0$, plot the parabola $y = x^2$ (right half of the parabola).

• For $x < 0$, plot the line $y = -x - 2$.

• At $x = 0$, the value is $f(0) = 0$ (from the first piece).

Table: Summary of Fundamental Functions

The following table summarizes the domain, range, and monotonicity (increasing/decreasing behavior) of the fundamental functions discussed above.

Function	Domain	Range	Increasing	Decreasing	Constant
Constant ($f(x) = c$)	$(-\infty, +\infty)$	$\{c\}$	None	None	$(-\infty, +\infty)$
Identity ($f(x) = x$)	$(-\infty, +\infty)$	$(-\infty, +\infty)$	$(-\infty, +\infty)$	None	None
Absolute Value ($f(x) = |x|$)	$(-\infty, +\infty)$	$[0, +\infty)$	$(0, +\infty)$	$(-\infty, 0)$	None
Square Root ($f(x) = \sqrt{x}$)	$[0, +\infty)$	$[0, +\infty)$	$[0, +\infty)$	None	None
Cubic ($f(x) = x^3$)	$(-\infty, +\infty)$	$(-\infty, +\infty)$	$(-\infty, +\infty)$	None	None
Cubic Root ($f(x) = \sqrt[3]{x}$)	$(-\infty, +\infty)$	$(-\infty, +\infty)$	$(-\infty, +\infty)$	None	None