Vocabulary Terms - Define/Illustrate

Question

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 $(0,\mathrm{\infty })$. 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 $(-\mathrm{\infty },0)$. 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 $2<x\le 5$.

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.