i need a definition of all of these words
arithmetic sequence
common difference
continuous graph
dependent variable
discrete graph
domain
function
function notation
independent variable
input
linear function
nonlinear function
output
range
relation
sequence
term of a sequence
vertical line test
Here is a definition of each term you've asked for:
1. Arithmetic Sequence:
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference. An example of an arithmetic sequence is 2, 5, 8, 11,... where the common difference is 3.
2. Common Difference:
In an arithmetic sequence, the common difference is the amount by which each term in the sequence differs from the previous term. It is a fixed number that, when added to any term of the sequence, gives the next term.
3. Continuous Graph:
A continuous graph is one where the points on the graph are connected and there are no breaks, gaps, or holes. The graph represents a function that has a value for every point within a certain range.
4. Dependent Variable:
In an experiment or mathematical function, the dependent variable is the variable that changes in response to changes in the independent variable. It is sometimes referred to as the "output" variable.
5. Discrete Graph:
A discrete graph is one where the points are distinct and unconnected, representing a function whose domain is a set of isolated points, often integers or whole numbers.
6. Domain:
The domain of a function is the set of all possible input values (usually "x" values) for which the function is defined.
7. Function:
A function is a relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output. It typically expresses a dependence of one quantity on another (e.g., f(x) = x^2 indicates that y depends on x).
8. Function Notation:
Function notation is a way to name a function that is defined by an equation. It is typically denoted by symbols such as f(x), g(x), h(x), etc., where "f" is the function name and "x" is the input variable.
9. Independent Variable:
The independent variable is the variable in an experiment or function that is manipulated or changed by the researcher to investigate its effect on the dependent variable. It is sometimes referred to as the "input" variable.
10. Input:
In the context of a function, the input is the value you put into the function. In the function f(x), "x" represents the input value.
11. Linear Function:
A linear function is a function whose graph forms a straight line. It can be represented by the equation y = mx + b, where m is the slope and b is the y-intercept.
12. Nonlinear Function:
A nonlinear function is any function that is not a linear function. Its graph does not form a straight line and can take many shapes, such as curves, parabolas, hyperbolas, etc.
13. Output:
In the context of a function, the output is the value you get out after applying the function to an input. For the function f(x), "f(x)" represents the output value when "x" is the input.
14. Range:
The range of a function is the set of all possible output values (usually "y" values) the function can produce.
15. Relation:
A relation is a set of ordered pairs, where pairs consist of an input (or element from the domain) and an output (or element from the codomain). A relation does not require each input to be associated with exactly one output, which is what differentiates a general relation from a function.
16. Sequence:
A sequence is an ordered list of numbers, which may or may not follow a specific rule or pattern. Each number in the list is called a term.
17. Term of a Sequence:
Each number in a sequence is known as a term. Terms are usually denoted by a subscript that represents their position in the sequence, for example, a_n represents the nth term of the sequence.
18. Vertical Line Test:
The vertical line test is a method used to determine if a graph represents a function. If any vertical line drawn on the graph intersects the graph at more than one point, then the graph does not represent a function. This is because a function can only have one output value for each input value.