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
