Sure! Let's break down each of the vocabulary terms you mentioned.
1. Increasing vs. Decreasing
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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.
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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
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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 \( (2, 5] \).
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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
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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.
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