Functions Lecture

Intro to Functions

MTHR001​

By Elo Cheng

Learning Objectives

• Determine whether a relation represents a function.

• Find teh value of a function.

• Determine whether a function is one-to-one.

• Use the vertical line test to identify functions.

  Identify the graphs listed in the library of functions.​

-a set of ordered pairs.

-The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. ​

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Relation

*Note that each value in the domain is also known as an input value, or independent variable while each value in the range is known as an output value, or dependent variable.

-a relation in which each possible input value leads to exactly one output value.

-it assigns a single value in the range to eah value in the domain. In other words, no x-values are repeated. ​

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Function

Which of the following is/are functions?

Multiple Choice

For the function {(0,1), (1,-3), (2,-4), (-4,1)}, which of the following is the correct domain and range?

• D: {1, -3, -4,}
  R: {0, 1, 2, -4}
• D:{0, 1, 2, -4}
  R:{1, -3, -4}
• D:{0, 1, 2, 3, 4}
  R:{1, -3, -4}

Multiple Choice

Is this mapping a function or not a function? 

• Function
• Not a Function 

Multiple Choice

Is this mapping a function or not a function? 

• Function
• Not a Function

Multiple Choice

Is the relation a function? Why.

• Yes, because the x-value 11 has two y-values pair with it.
• Yes, because each x-value has only one y-value paired with it.
• No, because the x-value 11 has two y-values pair with it.
• No, because each x-value has only one y-value paired with it.

The notation y=f(x) defines a function named f. This is read as "y is a function of x". In this case, "x" represents the input value while "f(x)" or "y" represents the output value.

Function Notation

A function N=f(y) gives the number of police officers, N, in a town in year y. What does f(2005)=300 represent?

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Interpreting Function Notation

Interpreting Function Notation

Evaluating Functions

Evaluating Functions

Evaluating Functions

Draw

Given the function $$h\left(p\right)=p^2+2p$$  , evaluate h(4).

Solving Functions

Draw

Given the function $$h\left(p\right)=p^2+2p$$  , solve for $$h\left(p\right)=8$$  

Evaluating and Solving a Tabular Function

Using the table, evaluate g(3) and solve g(n)=6.

Reading Function Values from a Graph

Given the graph, evaluate f(2) and solve f(x)=4.

One-to-one Function

-a function in which each output value corresponds to exactly one input value.

Using the Vertical Line Test

-Can be used to determine whether a graph represents a function.

-​If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function.

Using the Horizontal Line Test

-Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test.

-If any horizontal line ​intersects the graph more tahn once, then the graph does not represent a one-to-one function.

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Multiple Choice

Is y a function of x?

• function

• not a function

Multiple Choice

Is y a function of x?

• function

• not a function

Multiple Choice

Is y a function of x?

• function

• not a function

Multiple Choice

Is y a function of x?

• function

• not a function

Multiple Choice

The relation in the graph is..

• a function but not a one-to-one function.

• a one-to-one function.

• not a function

Multiple Choice

The relation in the graph is..

• a function but not a one-to-one function.

• a one-to-one function.

• not a function

Multiple Choice

The relation in the graph is..

• a function but not a one-to-one function.

• a one-to-one function.

• not a function

Multiple Choice

The relation in the graph is..

• a function but not a one-to-one function.

• a one-to-one function.

• not a function

Identifying Basic Toolkit Functions

Identifying Basic Toolkit Functions

Identifying Basic Toolkit Functions

Access the following online resources for additional instruction and practice with functions.

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• Determine if a Relation is a Function (http://openstaxcollege.org/l/relationfunction)

• Vertical Line Test (http://openstaxcollege.org/l/vertlinetest)

• Introduction to Functions (http://openstaxcollege.org/l/introtofunction)

• Vertical Line Test of Graph (http://openstaxcollege.org/l/vertlinegraph)

• One-to-one Functions (http://openstaxcollege.org/l/onetoone)

• Graphs as One-to-one Functions (http://openstaxcollege.org/l/graphonetoone)