Chapter 1.6: A Library of Functions – Graphs and Properties

Graphs and Functions

Introduction to a Library of Functions

This section introduces fundamental types of functions commonly encountered in algebra and precalculus. Understanding their definitions, properties, and graphs is essential for analyzing more complex mathematical models.

Linear Functions

Definition and Properties

• Linear Function: A function of the form $f(x) = mx + b$, where m and b are real numbers.

• Constant Function: If $m = 0$, then $f(x) = b$ is a constant function.

• Identity Function: If $m = 1$ and $b = 0$, then $f(x) = x$ is the identity function.

• The graph of a linear function is a nonvertical straight line with slope m and y-intercept b.

Example: Writing a Linear Function

Find a linear function $g$ such that $g(1) = 4$ and $g(-3) = -2$.

• Find the slope: $m = \frac{4 - (-2)}{1 - (-3)} = \frac{6}{4} = \frac{3}{2}$

• Use point-slope form: $y - 4 = \frac{3}{2}(x - 1)$

• Simplify: $g(x) = \frac{3}{2}x + \frac{5}{2}$

Application: Linear Models

Linear functions are used to model real-world relationships. For example, the length of a "Megatooth" shark can be estimated from tooth height:

• Formula: $\text{Shark length} = 0.96 \times \text{height of tooth} - 0.22$

• For a tooth of 15.6 cm: $\text{Shark length} = 0.96 \times 15.6 - 0.22 = 14.8$ meters

Root Functions

Square Root Function

• Definition: $f(x) = \sqrt{x}$

• Domain: $[0, \infty)$

• Range: $[0, \infty)$

• Graph passes through points where $x$ is a perfect square.

Cube Root Function

• Definition: $f(x) = \sqrt[3]{x}$

• Domain and Range: $(-\infty, \infty)$

• Graph passes through points where $x$ is a perfect cube.

Piecewise Functions

Definition and Evaluation

A piecewise function uses different formulas for different parts of its domain. To evaluate, determine which formula applies to the input value.

• Example: $F(x) = \begin{cases} x^2 & x < 1 \\ 2x + 1 & x \geq 1 \end{cases}$

• $F(0) = 0^2 = 0$ (since $0 < 1$)

• $F(2) = 2 \times 2 + 1 = 5$ (since $2 \geq 1$)

Application: Speeding Fines as a Piecewise Function

• For $x$ mph, $f(x) = \begin{cases} 0 & x \leq 55 \\ 50 + 3(x-55) & 56 \leq x < 75 \\ 150 + 5(x-75) & x \geq 75 \end{cases}$

• Fine for 60 mph: $f(60) = 50 + 3(60-55) = 65$

• Fine for 90 mph: $f(90) = 150 + 5(90-75) = 225$

Piecewise Functions from Data

Given points (1,1), (3,5), (5,2), connect with line segments and express as a piecewise function.

Step Functions

Greatest Integer Function

• Definition: $f(x) = \lfloor x \rfloor$ (the greatest integer less than or equal to $x$)

• Graph is a series of horizontal segments, jumping at each integer value.

Basic Functions and Their Properties

Summary Table of Common Functions

Function	Equation	Domain	Range	Symmetry
Constant	$f(x) = c$	$(-\infty, \infty)$	$\{c\}$	Even (y-axis)
Identity	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$	Odd (origin)
Squaring	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$	Even (y-axis)
Cubing	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$	Odd (origin)
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$[0, \infty)$	Even (y-axis)
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	None
Cube Root	$f(x) = \sqrt[3]{x}$	$(-\infty, \infty)$	$(-\infty, \infty)$	Odd (origin)
Reciprocal	$f(x) = \frac{1}{x}$	$(-\infty, 0) \cup (0, \infty)$	$(-\infty, 0) \cup (0, \infty)$	Odd (origin)
Reciprocal Square	$f(x) = \frac{1}{x^2}$	$(-\infty, 0) \cup (0, \infty)$	$(0, \infty)$	Even (y-axis)
Greatest Integer	$f(x) = \lfloor x \rfloor$	$(-\infty, \infty)$	Integers	None

Graphs of Basic Functions

• Identity Function: $f(x) = x$

• Squaring Function: $f(x) = x^2$

• Cubing Function: $f(x) = x^3$

• Absolute Value Function: $f(x) = |x|$

• Square Root Function: $f(x) = \sqrt{x}$

• Cube Root Function: $f(x) = \sqrt[3]{x}$

• Reciprocal Function: $f(x) = \frac{1}{x}$

• Reciprocal Square Function: $f(x) = \frac{1}{x^2}$

• Greatest Integer Function: $f(x) = \lfloor x \rfloor$

Additional info: Rational power functions and other variations can be constructed by combining these basic forms. Understanding their domains, ranges, and symmetries is crucial for graphing and analyzing more complex functions.