Chapter 1 Functions and their graphs

1.3Graphs of Functions part 1

Evaluate the following functions for f(-2), f(1) and f(3)

1. 2.

Evaluate the following Piecewise function for f(-1),f(0)

3.

Warm-up

1.f(-2)=2 f(1)=5 f(3)=27

2. f(-2)=-14 f(1)=1 f(3)=11 3. f(-1)=1 f(0)=-3

Warm-up answers

Students will be able to : *find the domain and ranges of functions

and use the vertical line test for functions. *Determine intervals on which functions are

increasing,decreasing,or constant. * Determine relative maximum and relative

minimum values of functions. Identify even and odd functions.

Objectives

The graph of the function f is the collection of ordered pairs (x,f(x)) such that x is in the domain of f.

What is Domain? Answer: Is the set of all possible values for x What is Range? Answer: Is the set of all possible values for y Example 1 show us how to use the graph od

the function to fund the domain and range.

The graph of a function

Example #1: Use the graph to find (a) the domain, (b) the range.

a)Domain: [-2,2] b)Range: (-

Finding the domain and range

𝑓 (𝑥 )=2− 12𝑥2

Lets say we have another graph What would be the domain ? Answer: [-4,What about the range?Answer: [-3,3]

Example #2

Find the range of the following figure:

Answer: [-1,1]

Example #3

Lets say instead of the graph we are given the function. How can we find the domain and range

To find the domain we need to solve for x we make it greater since square roots do

not take negatives We solve for x

+4 +4 that is my domain

Example #4

To find the range we can also look at the graph

Example#4 continue

Find the domain and range of the following function by graphing:

Look at graph Domain: Range:

Example #5Problem 11from book

Example Find the domain and range of the following

function graphically.

Solution:Domain: Range: set of all nonnegative real numbers

Student Practice

Do problems 12 and 13 from book

Student practice

What is the vertical line test? Answer: Is a test use in mathematics to

decide whether a given graph represents a function or not.

How does it works? Answer: basically, in order for a graph to be a

function a vertical line can only touch one point each time in the graph. If a vertical line touches two or more points in the graph at a time, then the graph does not represent a function.

Vertical Line Test

Lets see if the graph represents a function or not. Example #1

Lets see how a vertical line test works

Answer for Ex.1

Does the graph represents a function?

Example #2

Its not a function

Answer to Example #2

Does the graph represents a function?

Example #3

It’s a function even do it touches two points one of them does not exit.

Answer to Ex.3

Increasing and decreasing FunctionsHow do you know when a function is increasing or decreasing ? Increasing Functions A function is "increasing" if the y-value increases as the

x-value increases, like this:

It is easy to see that y=f(x) tends to go up as it goes along.

For a function to increase in the interval

Increasing and decreasing functions

when x1 < x2 then f(x1) ≤ f(x2)

Increasing

when x1 < x2 then f(x1) < f(x2)

Strictly Increasing

Increasing a decreasing functions Decreasing Functions The y-value decreases as the x-value

increases:

when x1 < x2 then f(x1) ≥ f(x2)

Decreasing

when x1 < x2 then f(x1) > f(x2)

Strictly Decreasing

A Constant Function is a horizontal line:

So

Constant functions

Examples of functions on which intervals does the functions increase, decrease?

Examples of increasing and decreasing functions

The quadratic function is decreasing on the interval and increasing on the interval

Examples solutions

On which intervals does the graph increase or decrease?

Solution: The cubic function is increasing in its entire domain

Examples

From book page 37 Problems # 7-9 From book page 38 Problems# 19-24

Homework

Today we saw about domain, range , vertical line test and about increasing and deceasing functions.

Tomorrow we are going to continue with the section with relative maxima and minimum and even and odd functions.

Closure