Graphing and Analyzing Basic Functions and Restricted Domains in College Algebra

Things to Know

• Determining the Domain of a Function Given the Equation: Understand how to find the domain of polynomial, rational, or root functions from their equations.

• Determining the Domain and Range of a Function from Its Graph: Learn to identify the domain and range by analyzing the graph of a function.

Objective 1: Sketching the Graphs of the Basic Functions

Introduction

Basic functions are foundational in algebra and their graphs reveal important properties. Memorizing these graphs and understanding their domains and ranges is essential for further study.

1. The Constant Function

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

• Graph: A horizontal line at $y = b$.

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: $\{b\}$ (only one value).

• Example: $f(x) = 3$ is a horizontal line at $y = 3$.

2. The Identity Function

• Definition: $f(x) = x$.

• Graph: A straight line through the origin with slope 1.

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

• Example: $f(2) = 2$.

3. The Square Function

• Definition: $f(x) = x^2$.

• Graph: A parabola opening upwards with vertex at (0,0).

• Domain: $(-\infty, \infty)$.

• Range: $[0, \infty)$.

• Example: $f(-3) = 9$.

4. The Cube Function

• Definition: $f(x) = x^3$.

• Graph: An S-shaped curve passing through the origin.

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

• Example: $f(2) = 8$.

5. The Square Root Function

• Definition: $f(x) = \sqrt{x}$.

• Graph: Starts at (0,0) and increases slowly to the right.

• Domain: $[0, \infty)$ (only non-negative $x$).

• Range: $[0, \infty)$.

• Example: $f(4) = 2$.

6. The Cube Root Function

• Definition: $f(x) = \sqrt[3]{x}$.

• Graph: Passes through the origin, similar to the cube function but less steep.

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

• Example: $f(-8) = -2$.

7. The Reciprocal Function

• Definition: $f(x) = \frac{1}{x}$.

• Graph: Two branches, one in the first quadrant and one in the third, with vertical and horizontal asymptotes at $x=0$ and $y=0$.

• Domain: $(-\infty, 0) \cup (0, \infty)$.

• Range: $(-\infty, 0) \cup (0, \infty)$.

• Example: $f(2) = 0.5$.

8. The Absolute Value Function

• Definition: $f(x) = |x|$.

• Graph: V-shaped, with vertex at (0,0).

• Domain: $(-\infty, \infty)$.

• Range: $[0, \infty)$.

• Piecewise Definition: $f(x) = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

9. The Greatest Integer Function

• Definition: $f(x) = \lfloor x \rfloor$ (also called the floor function).

• Graph: Step-like graph, jumps at each integer.

• Domain: $(-\infty, \infty)$.

• Range: All integers.

• Example: $f(2.5) = 2$.

Objective 2: Sketching the Graphs of Basic Functions With Restricted Domains

Introduction

Functions can be defined on restricted domains, which limits the set of input values. This affects the graph and properties of the function.

Restricted Domain Example: Square Function

• Function: $f(x) = x^2$ for $x > 0$.

• Domain: $(0, \infty)$.

• Graph: Only the right half of the parabola is shown.

Restricted Domain Example: Square Root Function

• Function: $f(x) = \sqrt{x}$ for $x \geq 1$.

• Domain: $[1, \infty)$.

• Graph: Starts at (1,1) and increases to the right.

Interval Notation and Inequality for Restricted Domains

• Interval Notation: Used to describe the set of $x$ values for which the function is defined. Example: $(-2, 1]$.

• Inequality: $-2 < x \leq 1$.

Piecewise-Defined Functions

• Definition: A function defined by different expressions on different intervals of its domain.

• Example: $f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 1 - x & \text{if } x \geq 1 \end{cases}$

• Graph: Each piece is graphed on its respective interval, with possible jumps or corners at the boundaries.

Evaluating and Graphing Piecewise Functions

• Example: For $f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 1 - x & \text{if } x \geq 1 \end{cases}$, evaluate $f(-3)$, $f(-1)$, $f(2)$:

  • $f(-3) = (-3)^2 = 9$

  • $f(-1) = (-1)^2 = 1$

  • $f(2) = 1 - 2 = -1$

• Domain: $(-\infty, \infty)$ (if both pieces cover all real numbers).

• Range: Determined by the outputs of each piece.

Graphing Absolute Value and Other Piecewise Functions

• Absolute Value Function: $f(x) = |x|$ is piecewise as shown above.

• Graph: V-shape, with two linear pieces meeting at the origin.

• Piecewise Example: For $f(x)$ defined as:

  • $f(x) = -3x - 6$ for $-2 < x < -2$

  • $f(x) = 2$ for $-2 \leq x \leq 1$

  • $f(x) = -x + 2$ for $x > 1$

  Each piece is graphed on its interval, with endpoints marked as open or closed depending on inclusion.

Domain and Range from Graphs

• Domain: The set of all $x$ values for which the function is defined. Example: $(-\infty, 0) \cup (0, \infty)$ for $f(x) = \frac{1}{x}$.

• Range: The set of all possible $y$ values. Example: $(-\infty, 0) \cup (0, \infty)$ for $f(x) = \frac{1}{x}$.

Table: Summary of Basic Functions

Function	Equation	Domain	Range	Graph Shape
Constant	$f(x) = b$	$(-\infty, \infty)$	$\{b\}$	Horizontal line
Identity	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$	Diagonal line
Square	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$	Parabola
Cube	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$	S-curve
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	Curve from origin
Cube Root	$f(x) = \sqrt[3]{x}$	$(-\infty, \infty)$	$(-\infty, \infty)$	S-curve
Reciprocal	$f(x) = \frac{1}{x}$	$(-\infty, 0) \cup (0, \infty)$	$(-\infty, 0) \cup (0, \infty)$	Hyperbola
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$[0, \infty)$	V-shape
Greatest Integer	$f(x) = \lfloor x \rfloor$	$(-\infty, \infty)$	Integers	Step function

Additional info:

• Piecewise functions are important for modeling situations where a rule changes depending on the input value.

• Restricted domains are used in applications such as physics, economics, and engineering, where only certain input values are meaningful.