Library of Functions and Piecewise-defined Functions: Study Notes

Section 1.4 – Library of Functions; Piecewise-defined Functions

Learning Objectives

• Determine when a mathematical relation is a function.

• Evaluate a function.

• Find the domain of a function.

• Find the difference quotient.

• Find the result of arithmetic operations with functions.

The Library of Functions

Introduction to Parent Functions

Parent functions are basic functions whose behavior and properties serve as foundational examples in algebra. Understanding these functions is essential for analyzing more complex functions and their transformations.

• Constant Function (Linear): $y = c$ - The graph is a horizontal line at $y = c$. - Domain: All real numbers ($\mathbb{R}$). - Range: $\{c\}$ (a single value).

• Identity Function (Linear): $y = x$ - The graph is a straight line passing through the origin with slope 1. - Domain and Range: All real numbers.

• Quadratic Function: $y = x^2$ - The graph is a parabola opening upwards. - Domain: All real numbers. - Range: $[0, \infty)$.

• Cubic Function: $y = x^3$ - The graph is an S-shaped curve passing through the origin. - Domain and Range: All real numbers.

• Square Root Function: $y = \sqrt{x}$ - The graph starts at the origin and increases slowly. - Domain: $[0, \infty)$. - Range: $[0, \infty)$.

• Reciprocal Function: $y = \frac{1}{x}$ - The graph has two branches, one in the first quadrant and one in the third, with a vertical and horizontal asymptote. - Domain: $x \neq 0$. - Range: $y \neq 0$.

• Absolute Value Function: $y = |x|$ - The graph is V-shaped, with the vertex at the origin. - Domain: All real numbers. - Range: $[0, \infty)$.

Table: Summary of Parent Functions

Function Name	Equation	Domain	Range	Graph Shape
Constant	$y = c$	All real numbers	$\{c\}$	Horizontal line
Identity	$y = x$	All real numbers	All real numbers	Diagonal line
Quadratic	$y = x^2$	All real numbers	$[0, \infty)$	Parabola
Cubic	$y = x^3$	All real numbers	All real numbers	S-curve
Square Root	$y = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	Curve starting at origin
Reciprocal	$y = \frac{1}{x}$	$x \neq 0$	$y \neq 0$	Hyperbola
Absolute Value	$y = |x|$	All real numbers	$[0, \infty)$	V-shape

Piecewise-defined Functions

Definition and Evaluation

A piecewise-defined function is a function that is defined by two or more equations, each corresponding to a specific part of the domain. These functions are useful for modeling situations where a rule changes depending on the input value.

• Definition: A function defined by multiple expressions, each valid for a certain interval of the domain.

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

• Evaluation: To evaluate $f(x)$ for a given $x$, determine which interval $x$ falls into and use the corresponding expression.

Example 1: Evaluating a Piecewise-defined Function

Given the function:

$f(x) = \begin{cases} 3x + 5 & \text{if } -4 \leq x < 1 \\ 4x - 2 & \text{if } x \geq 1 \end{cases}$

• To find $f(-7)$, $f(-4)$, $f(-2)$, $f(0)$, determine which piece applies for each $x$.

• For $x = -7$: $-7 < -4$, so $f(-7)$ is not defined by the given intervals.

• For $x = -4$: $-4 \leq x < 1$, so $f(-4) = 3(-4) + 5 = -12 + 5 = -7$.

• For $x = -2$: $-4 \leq x < 1$, so $f(-2) = 3(-2) + 5 = -6 + 5 = -1$.

• For $x = 0$: $-4 \leq x < 1$, so $f(0) = 3(0) + 5 = 5$.

• For $x \geq 1$, use $4x - 2$.

Graphing: Plot each piece on its respective interval, using the data from the table of values.

Domain: $[-4, \infty)$ (since the function is defined for $x \geq -4$).

Range: Determined by evaluating the outputs for each interval.

Intercepts: Find $x$ and $y$ intercepts by setting $f(x) = 0$ and $x = 0$ respectively.

Example 2: Another Piecewise-defined Function

Given the function:

$f(x) = \begin{cases} 3x + 1 & \text{if } x < 1 \\ -2x + 1 & \text{if } x \geq 1 \end{cases}$

• For $x = -3$: $x < 1$, so $f(-3) = 3(-3) + 1 = -9 + 1 = -8$.

• For $x = 0$: $x < 1$, so $f(0) = 3(0) + 1 = 1$.

• For $x \geq 1$, use $-2x + 1$.

Graphing: Plot each piece for its interval, using calculated values.

Domain: All real numbers (since both intervals together cover all $x$).

Range: Determined by evaluating outputs for each interval.

Intercepts: Find $x$ and $y$ intercepts as above.

Steps for Working with Piecewise-defined Functions

1. Identify the intervals and corresponding expressions.

2. For a given $x$, determine which interval it belongs to.

3. Evaluate the function using the appropriate expression.

4. Graph each piece on its interval, noting endpoints and possible discontinuities.

5. Determine domain, range, and intercepts from the graph.

Applications

• Modeling tax brackets, shipping rates, or any scenario where rules change at certain thresholds.

• Describing functions with abrupt changes, such as absolute value or step functions.

Additional info: The notes also reference difference quotients and arithmetic operations with functions, which are standard College Algebra topics. For completeness, students should review how to compute the difference quotient: $\frac{f(x+h) - f(x)}{h}$, and how to add, subtract, multiply, and divide functions.