Question

Vocabulary Terms - Define/Illustrate

Increasing vs Decreasing
Interval notation vs. Inequality notation
NonLinear functions

Answer 1

Sure! Let's break down each of the vocabulary terms you mentioned.

### 1. Increasing vs. Decreasing
- **Increasing Function**: A function \( f(x) \) is said to be increasing on an interval if, for any two points \( x_1 \) and \( x_2 \) in that interval, where \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \). This means that as you move from left to right on the graph, the function value goes up.

**Illustration**: An example is the function \( f(x) = x^2 \) on the interval \( (0, \infty) \). As \( x \) increases from \( 0 \), the function values also increase.

- **Decreasing Function**: A function \( f(x) \) is said to be decreasing on an interval if, for any two points \( x_1 \) and \( x_2 \) in that interval, where \( x_1 < x_2 \), it holds that \( f(x_1) > f(x_2) \). This means that as you move from left to right on the graph, the function value goes down.

**Illustration**: An example is the function \( f(x) = -x \) on the interval \( (-\infty, 0) \). As \( x \) increases from negative values towards zero, the function values decrease.

### 2. Interval Notation vs. Inequality Notation
- **Interval Notation**: This is a way of writing subsets of real numbers using parentheses and brackets.
- **Parentheses** (\( ( \) and \( ) \)) indicate that the endpoints are not included (open interval).
- **Brackets** (\( [ \) and \( ] \)) indicate that the endpoints are included (closed interval).

**Example**:
- The interval of all real numbers greater than 2 and less than or equal to 5 in interval notation is written as \( (2, 5] \).

- **Inequality Notation**: This expresses the same range of numbers using inequality symbols.

**Example**:
- Using inequality notation, the same interval would be expressed as \( 2 < x \leq 5 \).

### 3. Nonlinear Functions
- **Nonlinear Functions**: These are functions that do not form a straight line when graphed. In contrast to linear functions (which can be expressed in the form \( y = mx + b \)), nonlinear functions can take on a variety of forms, including quadratic functions, exponential functions, logarithmic functions, and trigonometric functions.

**Examples**:
- **Quadratic Function**: \( f(x) = x^2 \) is nonlinear because its graph is a parabola.
- **Exponential Function**: \( f(x) = 2^x \) is nonlinear and grows rapidly as \( x \) increases.
- **Trigonometric Function**: \( f(x) = \sin(x) \) is nonlinear, illustrating periodic behavior.

In summary, increasing and decreasing functions describe how the values of a function change with respect to its input, interval notation and inequality notation are two ways to express ranges of values, and nonlinear functions encompass a variety of functions that do not graph as straight lines.