Graph, Domain and Range of Arctan(x) functionSet the coefficients lớn ( a = 1, b = 1, c = 0 ) & ( d = 0 ) to obtain< f(x) = arctan(x) >Check that the domain of ( arctan(x) ) is the mix of all real numbers & the range is given by the interval ( (-dfracpi2 , dfracpi2 ) ). Check also that ( arctan(x) ) has horizontal asymptotes at ( y = -dfracpi2) and ( y = dfracpi2 ). (Zoom in & out)Change coefficient ( a ) and cảnh báo how the graph of ( a arctan(x) ) changes (vertical compression, stretching, reflection). How does it affect the range & the asymptotes of the function?Does a change in coefficient ( a ) affect the tên miền of the function?Change coefficient ( b ) and note how the graph of the function changes (horizontal compression, stretching). Does a change in ( b ) affect the range, domain and asymptotes of the function?Change coefficient ( c ) and chú ý how the graph of function changes (horizontal shift).

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Does a change ( c ) affect the domain, range & asymptotes of the function?Change coefficient ( d ) and note how the graph of the function changes (vertical shift). Does a change in ( d ) affect the range, domain & asymptotes of the function?If the range of ( arctan(x) ) is given by the interval ( (-dfracpi2 , dfracpi2 ) ) what is the range of ( a arctan(x) )? What is the range of ( a arctan(x) + d)?If the horizontal asymptotes of ( arctan(x) ) are given by the horizontal lines ( y = -dfracpi2 ) and ( y = dfracpi2 ) what are the horizontal asymptotes of ( a arctan(x) )?What are the horizontal asymptotes of ( a arctan(x) + d )?

## More References and liên kết to Inverse Trigonometric Functions

Inverse Trigonometric FunctionsGraph, Domain and Range of Arcsin functionGraph, Domain and Range of Arctan functionFind Domain & Range of Arccosine FunctionsFind Domain và Range of Arcsine FunctionsSolve Inverse Trigonometric Functions Questions

## Definition of arctan(x) Functions

Examining the graph of tan(x), shown below, we lưu ý that it is not a one to one function on its implied domain. But if we limit the domain to ( ( -dfracpi2 , dfracpi2 ) ), xanh graph below, we obtain a one lớn one function that has an inverse which cannot be obtained algebraically.  Example 1 Evaluate ( arctan(x) ) given the value of ( x ).Special values related to lớn special angles( arctan(0) = 0) because ( an(0) = 0 )( arctan(-1) = -dfracpi4 ; ext or -45^o ) because ( an(-dfracpi4) = -1 )( arctan(1) = dfracpi4 ; ext or 45^o) because ( an(dfracpi4) = 1 )Use of calculator( arctan(-2) = -1.107 ; ext or -63.43^o )( arctan(180) = 1.56 ; ext or 89.68^o )( arctan(-230) = -1.566 ; ext or -89.75^o )( arctan(-0.2) = -0.197 ; ext or -11.31^o )

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## Properties of ( y = arctan(x) )

Domain: ( (-infty , +infty) )Range: ( (-dfracpi2 , dfracpi2) )( arctan(-x) = - arctan(x) ) , odd function( arctan(x) ) is a one khổng lồ one function( an(arctan(x)) = x ) , due to property of a function & its inverse :( f(f^-1(x) = x ) where ( x ) is in the domain of ( f^-1 )( arctan( an(x)) = x ) , for x in the interval ( (-dfracpi2,dfracpi2) ) , due lớn property of a function and its inverse :( f^-1(f(x) = x ) where ( x ) is in the domain of ( f )Example 2Find the range of the functions:a) ( y = 3 arctan(x)) b) ( y = - arctan(x) + pi/2 ) c) ( y = 2 arctan(x + 3) - pi/4 )Solution lớn Example 2a)the range is found by first writing the range of ( arctan(x)) as a double inequality( -dfracpi2 lt arctan(x) lt dfracpi2 )multiply all terms of the above inequality by 3 và simplify( - 3 pi / 2 lt 3 arctan(x) lt 3 pi / 2 )the range of the given function ( y = 3 arctan(x) ) is given by the interval ( ( - 3 pi / 2 , 3 pi / 2 ) ).b)we start with the range of ( arctan(x))( -dfracpi2 lt arctan(x) lt dfracpi2 )multiply all terms of the above inequality by -1 and change symbol of the double inequality( -dfracpi2 lt -arctan(x) lt dfracpi2 )add ( dfracpi2 ) khổng lồ all terms of the double inequality above and simplify( 0 lt - arctan(x) + dfracpi2 lt pi )the range of the given function ( y = - arctan(x) + dfracpi2 ) is given by the interval ( ( 0, pi ) ).c)The graph of the function ( y = arctan(x+3)) is the graph of ( arctan(x)) shifted 3 unit to lớn the left. Shifting a graph to the left or to the right does not affect the range. Hencethe range of ( arctan(x+3)) is given by the double inequality( -dfracpi2 lt arctan(x+3) lt dfracpi2 )Multiply all terms of the double inequality by 2 and simplify( - pi lt arctan(x+3) lt pi )Add ( -pi/4 ) to lớn all terms of the above inequality & simplify to lớn obtain the range of the given function ( y = 2 arctan(x + 3) - pi/4 ) by the double inequality( - 5pi / 4 lt arctan(x+3) lt 3 pi / 4 )Example 3Evaluate if possiblea) ( an(arctan(-1.5))) b) ( arctan( an(dfracpi7) ) ) c) ( arctan( an(dfrac-pi2)) ) d) ( an(arctan(-19.5)) ) e) ( arctan( an(dfrac 13 pi6) ) )Solution to Example 3a)( an(arctan(-1.5)) = -1.5 ) using property 5 aboveb)( arctan( an(dfracpi7) = dfracpi7) using property 6 abovec)NOTE that we cannot use property 6 because ( an(dfrac-pi2) ) is undefined( arctan an(dfrac-pi2) ) is undefinedd)( an(arctan(-19.5)) = -19.5) use property 5e)Property 6 cannot be applied to the question in part e) because ( dfrac13pi4 ) is not in the domain name of that property. Hence we first calculate ( an(dfrac 13 pi4) )( an(dfrac 13 pi4) = an(dfrac 12 pi4+dfrac pi4) = an(3pi+dfrac pi4) = an(dfrac pi4) = 1 )We now substitute ( an(dfrac 13pi4) ) by 1 in the given expression và simplify( arctan( an(dfrac 13 pi6) ) = arctan(1) = dfracpi4 )

## Interactive Tutorial lớn Explore the Transformed arctan(x)

The exploration is carried out by analyzing the effects of the coefficients ( a, b, c) and ( d ) included in the more general arctan function given by< f(x) = a arctan(b x + c) + d >Change coefficients ( a, b, c) & ( d ) and click on the button "draw" in the left panel below. Zoom in và out for better viewing. Depending on which coefficients are changing, we expect the graph to lớn be transformed through vertical và horizontal compression, stretching, reflection and also vertical shifting.

 a = 1