Certainly! Here are definitions and illustrations for each of the vocabulary terms you mentioned:
1. Increasing vs. Decreasing
Definition:
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Increasing: A function is said to be increasing on an interval if, as you move from left to right along the x-axis, the y-values of the function output increase. Specifically, for any two points \(x_1\) and \(x_2\) in the interval where \(x_1 < x_2\), it must hold that \(f(x_1) < f(x_2)\).
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Decreasing: A function is said to be decreasing on an interval if, as you move from left to right along the x-axis, the y-values of the function output decrease. For any two points \(x_1\) and \(x_2\) in the interval where \(x_1 < x_2\), it must hold that \(f(x_1) > f(x_2)\).
Illustration:
- Increasing Example: \(f(x) = x^2\) on the interval \([0, \infty)\). As \(x\) increases, \(f(x)\) also increases.
- Decreasing Example: \(f(x) = -x\) on the interval \((-\infty, 0)\). As \(x\) increases, \(f(x)\) decreases.
2. Interval Notation vs. Inequality Notation
Definition:
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Interval Notation: A way to describe a set of numbers between two endpoints, using parentheses and brackets. Parentheses \( () \) indicate that an endpoint is not included (open interval), while brackets \([ ]\) indicate that an endpoint is included (closed interval).
Example: The interval \((2, 5]\) includes all numbers greater than 2 and up to and including 5.
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Inequality Notation: A way to represent the same set of numbers using inequalities, such as \(2 < x \leq 5\).
Illustration:
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Interval Notation:
\((2, 5]\) -
Inequality Notation:
\(2 < x \leq 5\)
3. Nonlinear Functions
Definition:
- Nonlinear functions are functions that do not form a straight line when graphed. This means that the relationship between the input and output variables cannot be represented by a linear equation. Nonlinear functions can include polynomials of degree greater than one, trigonometric functions, exponential functions, and logarithmic functions.
Examples:
- Quadratic Function: \(f(x) = x^2\) — This graph forms a parabola.
- Exponential Function: \(f(x) = e^x\) — This graph shows rapid growth.
- Sine Function: \(f(x) = \sin(x)\) — This graph oscillates up and down.
In summary, increasing and decreasing functions refer to the behavior of functions over specific intervals, while interval notation and inequality notation provide different ways to express ranges of values. Nonlinear functions represent relationships that do not create straight lines when plotted on a graph.
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