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f ( x ) = 3 x + 5

one-to-one

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For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

For the following exercises, graph the functions.

For the following exercises, use [link] to approximate the values.

Graph of a parabola.

If f ( x ) = −2 , then solve for x .

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If f ( x ) = 1 , then solve for x .

x = 1.8   or  or  x = 1.8

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For the following exercises, use the function h ( t ) = 16 t 2 + 80 t to find the values.

h ( 2 ) h ( 1 ) 2 1

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h ( a ) h ( 1 ) a 1

64 + 80 a 16 a 2 1 + a = 16 a + 64

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Domain and Range

For the following exercises, find the domain of each function, expressing answers using interval notation.

f ( x ) = x 3 x 2 4 x 12

( , 2 ) ( 2 , 6 ) ( 6 , )

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f ( x ) = x 6 x 4

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Graph this piecewise function: f ( x ) = { x + 1          x < 2 2 x 3     x 2

Graph of f(x).
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Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from x = 1  to  x = 2.

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Graph of a parabola.

increasing ( 2 , ) ; decreasing ( , 2 )

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Graph of a function.

increasing ( 3 , 1 ) ; constant ( , 3 ) ( 1 , )

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Find the local minimum of the function graphed in [link] .

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Find the local extrema for the function graphed in [link] .

local minimum ( 2 , 3 ) ; local maximum ( 1 , 3 )

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For the graph in [link] , the domain of the function is [ 3 , 3 ] . The range is [ 10 , 10 ] . Find the absolute minimum of the function on this interval.

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Find the absolute maximum of the function graphed in [link] .

Graph of a cubic function.

( 1.8 , 10 )

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Composition of Functions

For the following exercises, find ( f g ) ( x ) and ( g f ) ( x ) for each pair of functions.

f ( x ) = 4 x , g ( x ) = 4 x

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f ( x ) = 3 x + 2 , g ( x ) = 5 6 x

( f g ) ( x ) = 17 18 x ; ( g f ) ( x ) = 7 18 x

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f ( x ) = x 2 + 2 x , g ( x ) = 5 x + 1

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f ( x ) = x + 2 ,   g ( x ) = 1 x

( f g ) ( x ) = 1 x + 2 ; ( g f ) ( x ) = 1 x + 2

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f ( x ) = x + 3 2 ,   g ( x ) = 1 x

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For the following exercises, find ( f g ) and the domain for ( f g ) ( x ) for each pair of functions.

f ( x ) = x + 1 x + 4 ,   g ( x ) = 1 x

( f g ) ( x ) = 1 + x 1 + 4 x ,   x 0 ,   x 1 4

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f ( x ) = 1 x + 3 ,   g ( x ) = 1 x 9

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f ( x ) = 1 x ,   g ( x ) = x

( f g ) ( x ) = 1 x , x > 0

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f ( x ) = 1 x 2 1 ,   g ( x ) = x + 1

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For the following exercises, express each function H as a composition of two functions f and g where H ( x ) = ( f g ) ( x ) .

H ( x ) = 2 x 1 3 x + 4

sample: g ( x ) = 2 x 1 3 x + 4 ; f ( x ) = x

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H ( x ) = 1 ( 3 x 2 4 ) 3

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Transformation of Functions

For the following exercises, sketch a graph of the given function.

f ( x ) = 4 [ | x 2 | 6 ]

Graph of f(x)
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f ( x ) = ( x + 2 ) 2 1

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For the following exercises, sketch the graph of the function g if the graph of the function f is shown in [link] .

Graph of f(x)

For the following exercises, write the equation for the standard function represented by each of the graphs below.

For the following exercises, determine whether each function below is even, odd, or neither.

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

Absolute Value Functions

For the following exercises, write an equation for the transformation of f ( x ) = | x | .

Graph of f(x).

f ( x ) = 1 2 | x + 2 | + 1

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Graph of f(x).

f ( x ) = 3 | x 3 | + 3

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For the following exercises, graph the absolute value function.

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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