Library of Functions and Piecewise-defined Functions: Study Notes for Precalculus

Library of Functions

Constant Function

The constant function is defined as $f(x) = b$, where b is a real number. This function assigns the same value to every input.

• Domain: All real numbers

• Range: The single value {b}

• Graph: A horizontal line at y = b

• Y-intercept: (0, b)

• Even function: Yes, since $f(-x) = f(x)$

Identity Function

The identity function is defined as $f(x) = x$. It is a linear function where the output equals the input.

• Domain and Range: All real numbers

• Graph: A line with slope 1 passing through the origin

• Y-intercept: (0, 0)

• Odd function: Yes, since $f(-x) = -f(x)$

• Increasing: Over its entire domain

Square Function

The square function is defined as $f(x) = x^2$. Its graph is a parabola opening upwards.

• Domain: All real numbers

• Range: All non-negative real numbers ($[0, \infty)$)

• Intercept: (0, 0)

• Even function: Yes, since $f(-x) = f(x)$

• Decreasing: On $(-\infty, 0)$

• Increasing: On $(0, \infty)$

Cube Function

The cube function is defined as $f(x) = x^3$. Its graph is an S-shaped curve passing through the origin.

• Domain and Range: All real numbers

• Intercept: (0, 0)

• Odd function: Yes, since $f(-x) = -f(x)$

• Increasing: Over its entire domain

Square Root Function

The square root function is defined as $f(x) = \sqrt{x}$. Its graph starts at the origin and increases slowly.

• Domain: $[0, \infty)$ (non-negative real numbers)

• Range: $[0, \infty)$

• Intercept: (0, 0)

• Neither even nor odd: The function does not exhibit symmetry about the y-axis or origin.

• Increasing: On $[0, \infty)$

• Absolute minimum: 0 at $x = 0$

Cube Root Function

The cube root function is defined as $f(x) = \sqrt[3]{x}$. Its graph passes through the origin and is symmetric about the origin.

• Domain and Range: All real numbers

• Intercept: (0, 0)

• Odd function: Yes, since $f(-x) = -f(x)$

• Increasing: Over its entire domain

• No absolute extrema: The function does not have absolute maximum or minimum values.

Reciprocal Function

The reciprocal function is defined as $f(x) = \frac{1}{x}$. Its graph consists of two branches, one in each quadrant, and has no intercepts.

• Domain: All real numbers except $x = 0$

• Range: All real numbers except $y = 0$

• No intercepts: The graph does not cross the axes

• Odd function: Yes, since $f(-x) = -f(x)$

• Decreasing: On $(-\infty, 0)$ and $(0, \infty)$

Absolute Value Function

The absolute value function is defined as $f(x) = |x|$. Its graph is a V-shaped curve symmetric about the y-axis.

• Domain: All real numbers

• Range: $[0, \infty)$

• Intercept: (0, 0)

• Even function: Yes, since $f(-x) = f(x)$

• Decreasing: On $(-\infty, 0)$

• Increasing: On $(0, \infty)$

• Absolute minimum: 0 at $x = 0$

Greatest Integer Function (Step Function)

The greatest integer function, also known as the step function, is defined as $f(x) = \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to x.

• Domain: All real numbers

• Range: All integers

• Y-intercept: (0, 0)

• X-intercepts: In the interval [0, 1)

• Neither even nor odd: The function does not exhibit symmetry

• Constant: On every interval of the form [k, k+1), for integer k

Piecewise-defined Functions

Definition and Properties

A piecewise-defined function is a function defined by different expressions for different intervals of its domain. For example, the absolute value function can be written as:

• $f(x) = x$ if $x \geq 0$

• $f(x) = -x$ if $x < 0$

Piecewise-defined functions are useful for modeling situations where a rule changes depending on the input value.

Example: Application of Piecewise-defined Function

A trucking company charges per pound as follows:

• $0.50$ per mile for the first $100$ miles

• $0.40$ per mile for the next $300$ miles

• $0.20$ per mile for any distance in excess of $400$ miles

If $C$ is the charge per pound for hauling $x$ miles, then:

• For $0 < x \leq 100$: $C = 0.50x$

• For $100 < x \leq 400$: $C = 0.50 \times 100 + 0.40(x - 100)$

• For $x > 400$: $C = 0.50 \times 100 + 0.40 \times 300 + 0.20(x - 400)$

This is a classic example of a piecewise-defined function used in real-world applications.

Summary Table: Library of Functions

Function	Equation	Domain	Range	Symmetry	Intercepts
Constant	$f(x) = b$	All real numbers	{b}	Even	(0, b)
Identity	$f(x) = x$	All real numbers	All real numbers	Odd	(0, 0)
Square	$f(x) = x^2$	All real numbers	$[0, \infty)$	Even	(0, 0)
Cube	$f(x) = x^3$	All real numbers	All real numbers	Odd	(0, 0)
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	Neither	(0, 0)
Cube Root	$f(x) = \sqrt[3]{x}$	All real numbers	All real numbers	Odd	(0, 0)
Reciprocal	$f(x) = \frac{1}{x}$	$x \neq 0$	$y \neq 0$	Odd	None
Absolute Value	$f(x) = |x|$	All real numbers	$[0, \infty)$	Even	(0, 0)
Greatest Integer	$f(x) = \lfloor x \rfloor$	All real numbers	Integers	Neither	(0, 0)