Basics of functions and their graphs

Сентябрь 14, 2022

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Basics of functions and their graphs

Содержание

2. Find the domain and range of a relation. Determine whether a relation is a function. Determine
3. Definition of a Relation A relation is any set of ordered pairs. The set of all
4. Example: Finding the Domain and Range of a Relation Find the domain and range of the
5. Definition of a Function A function is a correspondence from a first set, called the domain,
6. Example: Determining Whether a Relation is a Function Determine whether the relation is a function: {(1,
7. Functions as Equations If an equation is solved for y and more than one value of
8. Example: Determining Whether an Equation Represents a Function Determine whether the equation defines y as a
9. Function Notation The special notation f(x), read “f of x” or “f at x”, represents the
10. Example: Evaluating a Function If evaluate Thus,
11. Graphs of Functions The graph of a function is the graph of its ordered pairs.
12. Example: Graphing Functions Graph the functions f(x) = 2x and g(x) = 2x – 3 in
13. Example: Graphing Functions (continued) We set up a partial table of coordinates for each function. We
14. The Vertical Line Test for Functions If any vertical line intersects a graph in more than
15. Example: Using the Vertical Line Test Use the vertical line test to identify graphs in which
16. Example: Analyzing the Graph of a Function Use the graph to find f(5) For what value
17. Identifying Domain and Range from a Function’s Graph To find the domain of a function from
18. Example: Identifying the Domain and Range of a Function from Its Graph Use the graph of
19. Example: Identifying the Domain and Range of a Function from Its Graph Use the graph of
20. Identifying Intercepts from a Function’s Graph To find the x-intercepts, look for the points at which
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Find the domain and range of a relation.
Determine whether a relation

is a function.
Determine whether an equation represents a function.
Evaluate a function.
Graph functions by plotting points.
Use the vertical line test to identify functions.
Obtain information about a function from its graph.
Identify the domain and range of a function from its graph.
Identify intercepts from a function’s graph.

Objectives:

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Definition of a Relation
A relation is any set of ordered pairs.

The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.

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Example: Finding the Domain and Range of a Relation
Find the domain

and range of the relation:
{(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (40, 21.2)}
domain: {0, 10, 20, 30, 40}
range: {9.1, 6.7, 10.7, 13.2, 21.2}

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Definition of a Function
A function is a correspondence from a first

set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element in the range.

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Example: Determining Whether a Relation is a Function
Determine whether the relation

is a function:
{(1, 2), (3, 4), (6, 5), (8, 5)}
No two ordered pairs in the given relation have the same first component and different second components. Thus, the relation is a function.

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Functions as Equations
If an equation is solved for y and more

than one value of y can be obtained for a given x, then the equation does not define y as a function of x.

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Example: Determining Whether an Equation Represents a Function
Determine whether the equation

defines y as a function of x.
The shows that for certain values of x, there are two values of y. For this reason, the equation does not define y as a function of x.

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Function Notation
The special notation f(x), read “f of x” or “f

at x”, represents the value of the function at the number x.

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Example: Evaluating a Function
If evaluate
Thus,

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Graphs of Functions
The graph of a function is the graph of

its ordered pairs.

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Example: Graphing Functions
Graph the functions f(x) = 2x and g(x) =

2x – 3 in the same rectangular coordinate system. Select integers for x, starting with –2 and ending with 2.

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Example: Graphing Functions (continued)
We set up a partial table of coordinates

for each function.

We then plot the points and connect them.

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The Vertical Line Test for Functions
If any vertical line intersects a

graph in more than one point, the graph does not define y as a function of x.

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Example: Using the Vertical Line Test
Use the vertical line test to

identify graphs in which y is a function of x.

not a function

function

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Example: Analyzing the Graph of a Function
Use the graph to

find f(5)

For what value of x
is f(x) = 100?
f(9) = 125, so x = 9.

f(5)=400

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Identifying Domain and Range from a Function’s Graph
To find the domain

of a function from it’s graph, look for all the inputs on the x-axis that correspond to points on the graph.
To find the range of a function from it’s graph, look for all the outputs on the y-axis that correspond to points on the graph.

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Example: Identifying the Domain and Range of a Function from Its

Graph

Use the graph of the function to identify its domain and its range.
Domain
Range

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Example: Identifying the Domain and Range of a Function from Its

Graph

Use the graph of the function to identify its domain and its range.
Domain
Range

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Identifying Intercepts from a Function’s Graph
To find the x-intercepts, look for

the points at which the graph crosses the x-axis.
To find the y-intercept, look for the point at which the graph crosses the y-axis.
A function can have more than one x-intercept but at most one y-intercept.