1.3graphs of functions part 1. 1.f(-2)=2 f(1)=5 f(3)=27 2. f(-2)=-14 f(1)=1 f(3)=11 3....
TRANSCRIPT
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Chapter 1 Functions and their graphs
1.3Graphs of Functions part 1
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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
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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
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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
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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
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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
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Lets say we have another graph What would be the domain ? Answer: [-4,What about the range?Answer: [-3,3]
Example #2
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Find the range of the following figure:
Answer: [-1,1]
Example #3
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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
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To find the range we can also look at the graph
Example#4 continue
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Find the domain and range of the following function by graphing:
Look at graph Domain: Range:
Example #5Problem 11from book
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Example Find the domain and range of the following
function graphically.
Solution:Domain: Range: set of all nonnegative real numbers
Student Practice
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Do problems 12 and 13 from book
Student practice
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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
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Lets see if the graph represents a function or not. Example #1
Lets see how a vertical line test works
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Answer for Ex.1
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Does the graph represents a function?
Example #2
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Its not a function
Answer to Example #2
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Does the graph represents a function?
Example #3
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It’s a function even do it touches two points one of them does not exit.
Answer to Ex.3
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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.
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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
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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
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A Constant Function is a horizontal line:
So
Constant functions
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Examples of functions on which intervals does the functions increase, decrease?
Examples of increasing and decreasing functions
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The quadratic function is decreasing on the interval and increasing on the interval
Examples solutions
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On which intervals does the graph increase or decrease?
Solution: The cubic function is increasing in its entire domain
Examples
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From book page 37 Problems # 7-9 From book page 38 Problems# 19-24
Homework
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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